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Inner product space

In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in $$\langle a, b \rangle$$. Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or scalar product of Cartesian coordinates. Inner product spaces of infinite dimension are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898.

An inner product naturally induces an associated norm, (denoted $$|x|$$ and $$|y|$$ in the picture); so, every inner product space is a normed vector space. If this normed space is also complete (that is, a Banach space) then the inner product space is a Hilbert space. If an inner product space $H$ is not a Hilbert space, it can be extended by completion to a Hilbert space $$\overline{H}.$$ This means that $$H$$ is a linear subspace of $$\overline{H},$$ the inner product of $$H$$ is the restriction of that of $$\overline{H},$$ and $$H$$ is dense in $$\overline{H}$$ for the topology defined by the norm.

Definition
In this article, $F$ denotes a field that is either the real numbers $$\R,$$ or the complex numbers $$\Complex.$$ A scalar is thus an element of $F$. A bar over an expression representing a scalar denotes the complex conjugate of this scalar. A zero vector is denoted $$\mathbf 0$$ for distinguishing it from the scalar $0$.

An inner product space is a vector space $V$ over the field $F$ together with an inner product, that is, a map
 * $$ \langle \cdot, \cdot \rangle : V \times V \to F $$

that satisfies the following three properties for all vectors $$x,y,z\in V$$ and all scalars $a,b\in F$.

a = \overline{a} $ if and only if $$a$$ is real, conjugate symmetry implies that $$\langle x, x \rangle $$ is always a real number. If $F$ is $$\R$$, conjugate symmetry is just symmetry. \langle ax+by, z \rangle = a \langle x, z \rangle + b \langle y, z \rangle.$$ \langle x, x \rangle > 0 $$ (conjugate symmetry implies that $$\langle x, x \rangle$$ is real).
 * Conjugate symmetry: $$\langle x, y \rangle = \overline{\langle y, x \rangle}.$$ As $
 * Linearity in the first argument: $$
 * Positive-definiteness: if $$x$$ is not zero, then $$

If the positive-definiteness condition is replaced by merely requiring that $$\langle x, x \rangle \geq 0$$ for all $$x$$, then one obtains the definition of positive semi-definite Hermitian form. A positive semi-definite Hermitian form $$\langle \cdot, \cdot \rangle$$ is an inner product if and only if for all $$x$$, if $$\langle x, x \rangle = 0$$ then $$x = \mathbf 0$$.

Basic properties
In the following properties, which result almost immediately from the definition of an inner product, $x, y$ and $z$ are arbitrary vectors, and $a$ and $b$ are arbitrary scalars.
 * $$\langle \mathbf{0}, x \rangle=\langle x,\mathbf{0}\rangle=0.$$
 * $$ \langle x, x \rangle$$ is real and nonnegative.
 * $$\langle x, x \rangle = 0$$ if and only if $$x=\mathbf{0}.$$
 * $$\langle x, ay+bz \rangle= \overline a \langle x, y \rangle + \overline b \langle x, z \rangle.$$ This implies that an inner product is a sesquilinear form.
 * $$\langle x + y, x + y \rangle = \langle x, x \rangle + 2\operatorname{Re}(\langle x, y \rangle) + \langle y, y \rangle,$$ where $$\operatorname{Re}$$ denotes the real part of its argument.

Over $$\R$$, conjugate-symmetry reduces to symmetry, and sesquilinearity reduces to bilinearity. Hence an inner product on a real vector space is a positive-definite symmetric bilinear form. The binomial expansion of a square becomes
 * $$\langle x + y, x + y \rangle = \langle x, x \rangle + 2\langle x, y \rangle + \langle y, y \rangle .$$

Convention variant
Some authors, especially in physics and matrix algebra, prefer to define inner products and sesquilinear forms with linearity in the second argument rather than the first. Then the first argument becomes conjugate linear, rather than the second. Bra-ket notation in quantum mechanics also uses slightly different notation, i.e. $$ \langle \cdot | \cdot \rangle $$, where $$ \langle x | y \rangle := \left ( y, x \right ) $$.

Notation
Several notations are used for inner products, including $$ \langle \cdot, \cdot \rangle $$, $$ \left ( \cdot, \cdot \right ) $$, $$ \langle \cdot | \cdot \rangle $$ and $$ \left ( \cdot | \cdot \right ) $$, as well as the usual dot product.

Real and complex numbers
Among the simplest examples of inner product spaces are $$\R$$ and $$\Complex.$$ The real numbers $$\R$$ are a vector space over $$\R$$ that becomes an inner product space with arithmetic multiplication as its inner product: $$\langle x, y \rangle := x y \quad \text{ for } x, y \in \R.$$

The complex numbers $$\Complex$$ are a vector space over $$\Complex$$ that becomes an inner product space with the inner product $$\langle x, y \rangle := x \overline{y} \quad \text{ for } x, y \in \Complex.$$ Unlike with the real numbers, the assignment $$(x, y) \mapsto x y$$ does define a complex inner product on $$\Complex.$$

Euclidean vector space
More generally, the real $n$-space $$\R^n$$ with the dot product is an inner product space, an example of a Euclidean vector space. $$ \left\langle \begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix}, \begin{bmatrix} y_1 \\ \vdots \\ y_n \end{bmatrix} \right\rangle = x^\textsf{T} y = \sum_{i=1}^n x_i y_i = x_1 y_1 + \cdots + x_n y_n, $$ where $$x^{\operatorname{T}}$$ is the transpose of $$x.$$

A function $$\langle \,\cdot, \cdot\, \rangle : \R^n \times \R^n \to \R$$ is an inner product on $$\R^n$$ if and only if there exists a symmetric positive-definite matrix $$\mathbf{M}$$ such that $$\langle x, y \rangle = x^{\operatorname{T}} \mathbf{M} y$$ for all $$x, y \in \R^n.$$ If $$\mathbf{M}$$ is the identity matrix then $$\langle x, y \rangle = x^{\operatorname{T}} \mathbf{M} y$$ is the dot product. For another example, if $$n = 2$$ and $$\mathbf{M} = \begin{bmatrix} a & b \\ b & d \end{bmatrix}$$ is positive-definite (which happens if and only if $$\det \mathbf{M} = a d - b^2 > 0$$ and one/both diagonal elements are positive) then for any $$x := \left[x_1, x_2\right]^{\operatorname{T}}, y := \left[y_1, y_2\right]^{\operatorname{T}} \in \R^2,$$ $$\langle x, y \rangle
 * = x^{\operatorname{T}} \mathbf{M} y

= \left[x_1, x_2\right] \begin{bmatrix} a & b \\ b & d \end{bmatrix} \begin{bmatrix} y_1 \\ y_2 \end{bmatrix} = a x_1 y_1 + b x_1 y_2 + b x_2 y_1 + d x_2 y_2.$$ As mentioned earlier, every inner product on $$\R^2$$ is of this form (where $$b \in \R, a > 0$$ and $$d > 0$$ satisfy $$a d > b^2$$).

Complex coordinate space
The general form of an inner product on $$\Complex^n$$ is known as the Hermitian form and is given by $$\langle x, y \rangle = y^\dagger \mathbf{M} x = \overline{x^\dagger \mathbf{M} y},$$ where $$M$$ is any Hermitian positive-definite matrix and $$y^{\dagger}$$ is the conjugate transpose of $$y.$$ For the real case, this corresponds to the dot product of the results of directionally-different scaling of the two vectors, with positive scale factors and orthogonal directions of scaling. It is a weighted-sum version of the dot product with positive weights—up to an orthogonal transformation.

Hilbert space
The article on Hilbert spaces has several examples of inner product spaces, wherein the metric induced by the inner product yields a complete metric space. An example of an inner product space which induces an incomplete metric is the space $$C([a, b])$$ of continuous complex valued functions $$f$$ and $$g$$ on the interval $$[a, b].$$ The inner product is $$\langle f, g \rangle = \int_a^b f(t) \overline{g(t)} \, \mathrm{d}t.$$ This space is not complete; consider for example, for the interval $[−1, 1]$ the sequence of continuous "step" functions, $$\{ f_k \}_k,$$ defined by: $$f_k(t) = \begin{cases} 0 & t \in [-1, 0] \\ 1 & t \in \left[\tfrac{1}{k}, 1\right] \\ kt & t \in \left(0, \tfrac{1}{k}\right) \end{cases}$$

This sequence is a Cauchy sequence for the norm induced by the preceding inner product, which does not converge to a function.

Random variables
For real random variables $$X$$ and $$Y,$$ the expected value of their product $$\langle X, Y \rangle = \mathbb{E}[XY]$$ is an inner product. In this case, $$\langle X, X \rangle = 0$$ if and only if $$\mathbb{P}[X = 0] = 1$$ (that is, $$X = 0$$ almost surely), where $$\mathbb{P}$$ denotes the probability of the event. This definition of expectation as inner product can be extended to random vectors as well.

Complex matrices
The inner product for complex square matrices of the same size is the Frobenius inner product $$\langle A, B \rangle := \operatorname{tr}\left(AB^\dagger\right)$$. Since trace and transposition are linear and the conjugation is on the second matrix, it is a sesquilinear operator. We further get Hermitian symmetry by, $$\langle A, B \rangle = \operatorname{tr}\left(AB^\dagger\right) = \overline{\operatorname{tr}\left(BA^\dagger\right)} = \overline{\left\langle B,A \right\rangle}$$ Finally, since for $$A$$ nonzero, $$\langle A, A\rangle = \sum_{ij} \left|A_{ij}\right|^2 > 0 $$, we get that the Frobenius inner product is positive definite too, and so is an inner product.

Vector spaces with forms
On an inner product space, or more generally a vector space with a nondegenerate form (hence an isomorphism $$V \to V^*$$), vectors can be sent to covectors (in coordinates, via transpose), so that one can take the inner product and outer product of two vectors—not simply of a vector and a covector.

Norm properties
Every inner product space induces a norm, called its, that is defined by $$\|x\| = \sqrt{\langle x, x \rangle}.$$ With this norm, every inner product space becomes a normed vector space.

So, every general property of normed vector spaces applies to inner product spaces. In particular, one has the following properties:

Absolute homogeneity:$\ Triangle inequality:$\

Cauchy–Schwarz inequality:$ Parallelogram law:$\ Polarization identity:$\ Ptolemy's inequality:$\

Orthogonality
Orthogonality:Two vectors $x$ and $y$ are said to be, often written $x \perp y,$ if their inner product is zero, that is, if $\langle x, y \rangle = 0.$

This happens if and only if $\ Orthogonal complement:The orthogonal complement of a subset $C \subseteq V$ is the set $C^{\bot}$ of the vectors that are orthogonal to all elements of $C$; that is, $C^{\bot} := \{\,y \in V : \langle y, c \rangle = 0 \text{ for all } c \in C\,\}.$ This set $C^{\bot}$ is always a closed vector subspace of $V$ and if the closure $\operatorname{cl}_V C$ of $C$ in $V$ is a vector subspace then $\operatorname{cl}_V C = \left(C^{\bot}\right)^{\bot}.$ Pythagorean theorem:If $x$ and $y$ are orthogonal, then $\ Parseval's identity:An induction on the Pythagorean theorem yields: if $x_1, \ldots, x_n$ are pairwise orthogonal, then $\sum_{i=1}^n \ Angle:When $\langle x, y \rangle$ is a real number then the Cauchy–Schwarz inequality implies that $\frac{\langle x, y \rangle}{\

Real and complex parts of inner products
Suppose that $$\langle \cdot, \cdot \rangle$$ is an inner product on $$V$$ (so it is antilinear in its second argument). The polarization identity shows that the real part of the inner product is $$\operatorname{Re} \langle x, y \rangle = \frac{1}{4} \left(\|x + y\|^2 - \|x - y\|^2\right).$$

If $$V$$ is a real vector space then $$\langle x, y \rangle = \operatorname{Re} \langle x, y \rangle = \frac{1}{4} \left(\|x + y\|^2 - \|x - y\|^2\right)$$ and the imaginary part (also called the ) of $$\langle \cdot, \cdot \rangle$$ is always $$0.$$

Assume for the rest of this section that $$V$$ is a complex vector space. The polarization identity for complex vector spaces shows that
 * $$\begin{alignat}{4}

\langle x, \ y \rangle &= \frac{1}{4} \left(\|x + y\|^2 - \|x - y\|^2 + i\|x + iy\|^2 - i\|x - iy\|^2 \right) \\ &= \operatorname{Re} \langle x, y \rangle + i \operatorname{Re} \langle x, i y \rangle. \\ \end{alignat}$$

The map defined by $$\langle x \mid y \rangle = \langle y, x \rangle$$ for all $$x, y \in V$$ satisfies the axioms of the inner product except that it is antilinear in its, rather than its second, argument. The real part of both $$\langle x \mid y \rangle$$ and $$\langle x, y \rangle$$ are equal to $$\operatorname{Re} \langle x, y \rangle$$ but the inner products differ in their complex part:
 * $$\begin{alignat}{4}

\langle x \mid y \rangle &= \frac{1}{4} \left(\|x + y\|^2 - \|x - y\|^2 - i\|x + iy\|^2 + i\|x - iy\|^2 \right) \\ &= \operatorname{Re} \langle x, y \rangle - i \operatorname{Re} \langle x, i y \rangle. \\ \end{alignat}$$

The last equality is similar to the formula expressing a linear functional in terms of its real part.

These formulas show that every complex inner product is completely determined by its real part. Moreover, this real part defines an inner product on $$V,$$ considered as a real vector space. There is thus a one-to-one correspondence between complex inner products on a complex vector space $$V,$$ and real inner products on $$V.$$

For example, suppose that $$V = \Complex^n$$ for some integer $$n > 0.$$ When $$V$$ is considered as a real vector space in the usual way (meaning that it is identified with the $$2 n-$$dimensional real vector space $$\R^{2n},$$ with each $$\left(a_1 + i b_1, \ldots, a_n + i b_n\right) \in \Complex^n$$ identified with $$\left(a_1, b_1, \ldots, a_n, b_n\right) \in \R^{2n}$$), then the dot product $$x \,\cdot\, y = \left(x_1, \ldots, x_{2n}\right) \, \cdot \, \left(y_1, \ldots, y_{2n}\right) := x_1 y_1 + \cdots + x_{2n} y_{2n}$$ defines a real inner product on this space. The unique complex inner product $$\langle \,\cdot, \cdot\, \rangle$$ on $$V = \C^n$$ induced by the dot product is the map that sends $$c = \left(c_1, \ldots, c_n\right), d = \left(d_1, \ldots, d_n\right) \in \Complex^n$$ to $$\langle c, d \rangle := c_1 \overline{d_1} + \cdots + c_n \overline{d_n}$$ (because the real part of this map $$\langle \,\cdot, \cdot\, \rangle$$ is equal to the dot product).

Real vs. complex inner products

Let $$V_{\R}$$ denote $$V$$ considered as a vector space over the real numbers rather than complex numbers. The real part of the complex inner product $$\langle x, y \rangle$$ is the map $$\langle x, y \rangle_{\R} = \operatorname{Re} \langle x, y \rangle ~:~ V_{\R} \times V_{\R} \to \R,$$ which necessarily forms a real inner product on the real vector space $$V_{\R}.$$ Every inner product on a real vector space is a bilinear and symmetric map.

For example, if $$V = \Complex$$ with inner product $$\langle x, y \rangle = x \overline{y},$$ where $$V$$ is a vector space over the field $$\Complex,$$ then $$V_{\R} = \R^2$$ is a vector space over $$\R$$ and $$\langle x, y \rangle_{\R}$$ is the dot product $$x \cdot y,$$ where $$x = a + i b \in V = \Complex$$ is identified with the point $$(a, b) \in V_{\R} = \R^2$$ (and similarly for $$y$$); thus the standard inner product $$\langle x, y \rangle = x \overline{y},$$ on $$\Complex$$ is an "extension" the dot product. Also, had $$\langle x, y \rangle$$ been instead defined to be the $$ $$\langle x, y \rangle = x y$$ (rather than the usual $$ $$\langle x, y \rangle = x \overline{y}$$) then its real part $$\langle x, y \rangle_{\R}$$ would  be the dot product; furthermore, without the complex conjugate, if $$x \in \C$$ but $$x \not\in \R$$ then $$\langle x, x \rangle = x x = x^2 \not\in [0, \infty)$$ so the assignment $$x \mapsto \sqrt{\langle x, x \rangle}$$ would not define a norm.

The next examples show that although real and complex inner products have many properties and results in common, they are not entirely interchangeable. For instance, if $$\langle x, y \rangle = 0$$ then $$\langle x, y \rangle_{\R} = 0,$$ but the next example shows that the converse is in general true. Given any $$x \in V,$$ the vector $$i x$$ (which is the vector $$x$$ rotated by 90°) belongs to $$V$$ and so also belongs to $$V_{\R}$$ (although scalar multiplication of $$x$$ by $$i = \sqrt{-1}$$ is not defined in $$V_{\R},$$ the vector in $$V$$ denoted by $$i x$$ is nevertheless still also an element of $$V_{\R}$$). For the complex inner product, $$\langle x, ix \rangle = -i \|x\|^2,$$ whereas for the real inner product the value is always $$\langle x, ix \rangle_{\R} = 0.$$

If $$\langle \,\cdot, \cdot\, \rangle$$ is a complex inner product and $$A : V \to V$$ is a continuous linear operator that satisfies $$\langle x, A x \rangle = 0$$ for all $$x \in V,$$ then $$A = 0.$$ This statement is no longer true if $$\langle \,\cdot, \cdot\, \rangle$$ is instead a real inner product, as this next example shows. Suppose that $$V = \Complex$$ has the inner product $$\langle x, y \rangle := x \overline{y}$$ mentioned above. Then the map $$A : V \to V$$ defined by $$A x = ix$$ is a linear map (linear for both $$V$$ and $$V_{\R}$$) that denotes rotation by $$90^{\circ}$$ in the plane. Because $$x$$ and $$A x$$ are perpendicular vectors and $$\langle x, Ax \rangle_{\R}$$ is just the dot product, $$\langle x, Ax \rangle_{\R} = 0$$ for all vectors $$x;$$ nevertheless, this rotation map $$A$$ is certainly not identically $$0.$$ In contrast, using the complex inner product gives $$\langle x, Ax \rangle = -i \|x\|^2,$$ which (as expected) is not identically zero.

Orthonormal sequences
Let $$V$$ be a finite dimensional inner product space of dimension $$n.$$ Recall that every basis of $$V$$ consists of exactly $$n$$ linearly independent vectors. Using the Gram–Schmidt process we may start with an arbitrary basis and transform it into an orthonormal basis. That is, into a basis in which all the elements are orthogonal and have unit norm. In symbols, a basis $$\{e_1, \ldots, e_n\}$$ is orthonormal if $$\langle e_i, e_j \rangle = 0$$ for every $$i \neq j$$ and $$\langle e_i, e_i \rangle = \|e_a\|^2 = 1$$ for each index $$i.$$

This definition of orthonormal basis generalizes to the case of infinite-dimensional inner product spaces in the following way. Let $$V$$ be any inner product space. Then a collection $$E = \left\{ e_a \right\}_{a \in A}$$ is a for $$V$$ if the subspace of $$V$$ generated by finite linear combinations of elements of $$E$$ is dense in $$V$$ (in the norm induced by the inner product). Say that $$E$$ is an for $$V$$ if it is a basis and $$\left\langle e_{a}, e_{b} \right\rangle = 0$$ if $$a \neq b$$ and $$\langle e_a, e_a \rangle = \|e_a\|^2 = 1$$ for all $$a, b \in A.$$

Using an infinite-dimensional analog of the Gram-Schmidt process one may show:

Theorem. Any separable inner product space has an orthonormal basis.

Using the Hausdorff maximal principle and the fact that in a complete inner product space orthogonal projection onto linear subspaces is well-defined, one may also show that

Theorem. Any complete inner product space has an orthonormal basis.

The two previous theorems raise the question of whether all inner product spaces have an orthonormal basis. The answer, it turns out is negative. This is a non-trivial result, and is proved below. The following proof is taken from Halmos's A Hilbert Space Problem Book (see the references).

 * {| class="toccolours collapsible collapsed" width="90%" style="text-align:left"

!Proof
 * Recall that the dimension of an inner product space is the cardinality of a maximal orthonormal system that it contains (by Zorn's lemma it contains at least one, and any two have the same cardinality). An orthonormal basis is certainly a maximal orthonormal system but the converse need not hold in general. If $$G$$ is a dense subspace of an inner product space $$V,$$ then any orthonormal basis for $$G$$ is automatically an orthonormal basis for $$V.$$ Thus, it suffices to construct an inner product space $$V$$ with a dense subspace $$G$$ whose dimension is strictly smaller than that of $$V.$$
 * Recall that the dimension of an inner product space is the cardinality of a maximal orthonormal system that it contains (by Zorn's lemma it contains at least one, and any two have the same cardinality). An orthonormal basis is certainly a maximal orthonormal system but the converse need not hold in general. If $$G$$ is a dense subspace of an inner product space $$V,$$ then any orthonormal basis for $$G$$ is automatically an orthonormal basis for $$V.$$ Thus, it suffices to construct an inner product space $$V$$ with a dense subspace $$G$$ whose dimension is strictly smaller than that of $$V.$$

Let $$K$$ be a Hilbert space of dimension $\aleph_0.$ (for instance, $$K = \ell^2(\N)$$). Let $$E$$ be an orthonormal basis of $$K,$$ so $$|E| = \aleph_0.$$ Extend $$E$$ to a Hamel basis $$E \cup F$$ for $$K,$$where $$E \cap F = \varnothing.$$ Since it is known that the Hamel dimension of $$K$$ is $$c,$$ the cardinality of the continuum, it must be that $$|F| = c.$$

Let $$L$$ be a Hilbert space of dimension $$c$$ (for instance, $$L = \ell^2(\R)$$). Let $$B$$ be an orthonormal basis for $$L$$ and let $$\varphi : F \to B$$ be a bijection. Then there is a linear transformation $$T : K \to L$$ such that $$T f = \varphi(f)$$ for $$f \in F,$$ and $$Te = 0$$ for $$e \in E.$$

Let $$V = K \oplus L$$ and let $$G = \{ (k, T k) : k \in K \}$$ be the graph of $$T.$$ Let $$\overline{G}$$ be the closure of $$G$$ in $$V$$; we will show $$\overline{G} = V.$$ Since for any $$e \in E$$ we have $$(e, 0) \in G,$$ it follows that $$K \oplus 0 \subseteq \overline{G}.$$

Next, if $$b \in B,$$ then $$b = T f$$ for some $$f \in F \subseteq K,$$ so $$(f, b) \in G \subseteq \overline{G}$$; since $$(f, 0) \in \overline{G}$$ as well, we also have $$(0, b) \in \overline{G}.$$ It follows that $$0 \oplus L \subseteq \overline{G},$$ so $$\overline{G} = V,$$ and $$G$$ is dense in $$V.$$

Finally, $$\{(e, 0) : e \in E \}$$ is a maximal orthonormal set in $$G$$; if $$0 = \langle (e, 0), (k, Tk) \rangle = \langle e, k \rangle + \langle 0, Tk \rangle = \langle e, k \rangle$$ for all $$e \in E$$ then $$k = 0,$$ so $$(k, Tk) = (0, 0)$$ is the zero vector in $$G.$$ Hence the dimension of $$G$$ is $$|E| = \aleph_0,$$ whereas it is clear that the dimension of $$V$$ is $$c.$$ This completes the proof.
 * }

Parseval's identity leads immediately to the following theorem:

Theorem. Let $$V$$ be a separable inner product space and $$\left\{e_k\right\}_k$$ an orthonormal basis of $$V.$$ Then the map $$x \mapsto \bigl\{\langle e_k, x \rangle\bigr\}_{k \in \N}$$ is an isometric linear map $$V \rightarrow \ell^2$$ with a dense image.

This theorem can be regarded as an abstract form of Fourier series, in which an arbitrary orthonormal basis plays the role of the sequence of trigonometric polynomials. Note that the underlying index set can be taken to be any countable set (and in fact any set whatsoever, provided $$\ell^2$$ is defined appropriately, as is explained in the article Hilbert space). In particular, we obtain the following result in the theory of Fourier series:

Theorem. Let $$V$$ be the inner product space $$C[-\pi, \pi].$$ Then the sequence (indexed on set of all integers) of continuous functions $$e_k(t) = \frac{e^{i k t}}{\sqrt{2 \pi}}$$ is an orthonormal basis of the space $$C[-\pi, \pi]$$ with the $$L^2$$ inner product. The mapping $$f \mapsto \frac{1}{\sqrt{2 \pi}} \left\{\int_{-\pi}^\pi f(t) e^{-i k t} \, \mathrm{d}t \right\}_{k \in \Z}$$ is an isometric linear map with dense image.

Orthogonality of the sequence $$\{ e_k \}_k$$ follows immediately from the fact that if $$k \neq j,$$ then $$\int_{-\pi}^\pi e^{-i (j - k) t} \, \mathrm{d}t = 0.$$

Normality of the sequence is by design, that is, the coefficients are so chosen so that the norm comes out to 1. Finally the fact that the sequence has a dense algebraic span, in the, follows from the fact that the sequence has a dense algebraic span, this time in the space of continuous periodic functions on $$[-\pi, \pi]$$ with the uniform norm. This is the content of the Weierstrass theorem on the uniform density of trigonometric polynomials.

Operators on inner product spaces
Several types of linear maps $$A : V \to W$$ between inner product spaces $$V$$ and $$W$$ are of relevance:
 * : $$A : V \to W$$ is linear and continuous with respect to the metric defined above, or equivalently, $$A$$ is linear and the set of non-negative reals $$\{ \|Ax\| : \|x\| \leq 1\},$$ where $$x$$ ranges over the closed unit ball of $$V,$$ is bounded.
 * : $$A : V \to W$$ is linear and $$\langle Ax, y \rangle = \langle x, Ay \rangle$$ for all $$x, y \in V.$$
 * : $$A : V \to W$$ satisfies $$\|A x\| = \|x\|$$ for all $$x \in V.$$ A (resp. an ) is an isometry that is also a linear map (resp. an antilinear map). For inner product spaces, the polarization identity can be used to show that $$A$$ is an isometry if and only if $$\langle Ax, Ay \rangle = \langle x, y \rangle$$ for all $$x, y \in V.$$  All isometries are injective. The Mazur–Ulam theorem establishes that every surjective isometry between two  normed spaces is an affine transformation. Consequently, an isometry $$A$$ between real inner product spaces is a linear map if and only if $$A(0) = 0.$$ Isometries are morphisms between inner product spaces, and morphisms of real inner product spaces are orthogonal transformations (compare with orthogonal matrix).
 * : $$A : V \to W$$ is an isometry which is surjective (and hence bijective). Isometrical isomorphisms are also known as unitary operators (compare with unitary matrix).

From the point of view of inner product space theory, there is no need to distinguish between two spaces which are isometrically isomorphic. The spectral theorem provides a canonical form for symmetric, unitary and more generally normal operators on finite dimensional inner product spaces. A generalization of the spectral theorem holds for continuous normal operators in Hilbert spaces.

Generalizations
Any of the axioms of an inner product may be weakened, yielding generalized notions. The generalizations that are closest to inner products occur where bilinearity and conjugate symmetry are retained, but positive-definiteness is weakened.

Degenerate inner products
If $$V$$ is a vector space and $$\langle \,\cdot\,, \,\cdot\, \rangle$$ a semi-definite sesquilinear form, then the function: $$\|x\| = \sqrt{\langle x, x\rangle}$$ makes sense and satisfies all the properties of norm except that $$\|x\| = 0$$ does not imply $$x = 0$$ (such a functional is then called a semi-norm). We can produce an inner product space by considering the quotient $$W = V / \{x : \|x\| = 0\}.$$ The sesquilinear form $$\langle \,\cdot\,, \,\cdot\, \rangle$$ factors through $$W.$$

This construction is used in numerous contexts. The Gelfand–Naimark–Segal construction is a particularly important example of the use of this technique. Another example is the representation of semi-definite kernels on arbitrary sets.

Nondegenerate conjugate symmetric forms
Alternatively, one may require that the pairing be a nondegenerate form, meaning that for all non-zero $$x \neq 0$$ there exists some $$y$$ such that $$\langle x, y \rangle \neq 0,$$ though $$y$$ need not equal $$x$$; in other words, the induced map to the dual space $$V \to V^*$$ is injective. This generalization is important in differential geometry: a manifold whose tangent spaces have an inner product is a Riemannian manifold, while if this is related to nondegenerate conjugate symmetric form the manifold is a pseudo-Riemannian manifold. By Sylvester's law of inertia, just as every inner product is similar to the dot product with positive weights on a set of vectors, every nondegenerate conjugate symmetric form is similar to the dot product with weights on a set of vectors, and the number of positive and negative weights are called respectively the positive index and negative index. Product of vectors in Minkowski space is an example of indefinite inner product, although, technically speaking, it is not an inner product according to the standard definition above. Minkowski space has four dimensions and indices 3 and 1 (assignment of "+" and "−" to them differs depending on conventions).

Purely algebraic statements (ones that do not use positivity) usually only rely on the nondegeneracy (the injective homomorphism $$V \to V^*$$) and thus hold more generally.

Related products
The term "inner product" is opposed to outer product, which is a slightly more general opposite. Simply, in coordinates, the inner product is the product of a $$1 \times n$$ with an $$n \times 1$$ vector, yielding a $$1 \times 1$$ matrix (a scalar), while the outer product is the product of an $$m \times 1$$ vector with a $$1 \times n$$ covector, yielding an $$m \times n$$ matrix. The outer product is defined for different dimensions, while the inner product requires the same dimension. If the dimensions are the same, then the inner product is the of the outer product (trace only being properly defined for square matrices). In an informal summary: "inner is horizontal times vertical and shrinks down, outer is vertical times horizontal and expands out".

More abstractly, the outer product is the bilinear map $$W \times V^* \to \hom(V, W)$$ sending a vector and a covector to a rank 1 linear transformation (simple tensor of type (1, 1)), while the inner product is the bilinear evaluation map $$V^* \times V \to F$$ given by evaluating a covector on a vector; the order of the domain vector spaces here reflects the covector/vector distinction.

The inner product and outer product should not be confused with the interior product and exterior product, which are instead operations on vector fields and differential forms, or more generally on the exterior algebra.

As a further complication, in geometric algebra the inner product and the (Grassmann) product are combined in the geometric product (the Clifford product in a Clifford algebra) – the inner product sends two vectors (1-vectors) to a scalar (a 0-vector), while the exterior product sends two vectors to a bivector (2-vector) – and in this context the exterior product is usually called the  (alternatively, ). The inner product is more correctly called a product in this context, as the nondegenerate quadratic form in question need not be positive definite (need not be an inner product).

title: Inner product space


text:
Iain Banks

Iain Banks (16 February 1954 – 9 June 2013) was a Scottish author, writing mainstream fiction as Iain Banks and science fiction as Iain M. Banks, adding the initial of his adopted middle name Menzies. After the success of The Wasp Factory (1984), he began to write full time. His first science fiction book, Consider Phlebas, appeared in 1987, marking the start of the Culture series. His books have been adapted for theatre, radio, and television. In 2008, The Times named Banks in their list of "The 50 greatest British writers since 1945".

In April 2013, Banks announced he had inoperable cancer and was unlikely to live beyond a year. He died on 9 June 2013.

Early life
Banks was born in Dunfermline, Fife, to a mother who was a professional ice skater and a father who was an officer in the Admiralty. An only child, he lived in North Queensferry until the age of nine, near the naval dockyards in Rosyth, where his father was based. The family then moved to Gourock due to his father's work. When someone introduced him to science fiction by giving him Kemlo and the Zones of Silence by Reginald Alec Martin, he continued reading the series, which encouraged him to write science fiction himself. After attending Gourock and Greenock High Schools, Banks studied English, philosophy, and psychology at the University of Stirling (1972–1975).

After graduation, Banks took a succession of jobs that left him free to write in the evenings. These supported his writing throughout his twenties and allowed him to take long breaks between contracts, during which time he travelled through Europe and North America. During this period he worked as an IBM 'Expediter Analyser' (a kind of procurement clerk), a testing technician for the British Steel Corporation, and a costing clerk for a law firm in London's Chancery Lane.

Writing career
Banks took up writing at the age of 11. He completed a first novel, The Hungarian Lift-Jet, at 16 and a second, TTR (also entitled The Tashkent Rambler) in his first year at Stirling University in 1972. Though he saw himself mainly as a science fiction author, his publishing problems led him to pursue mainstream fiction. His first published novel The Wasp Factory, appeared in 1984, when he was thirty. After the success of The Wasp Factory, Banks began to write full time. His editor at Macmillan, James Hale, advised him to write a book a year, which he agreed to do.

His second novel Walking on Glass followed in 1985, then The Bridge in 1986, and in 1987 Espedair Street, which was later broadcast as a series on BBC Radio 4. His first published science fiction book, Consider Phlebas, emerged in 1987 and was the first of several in the acclaimed Culture series. Banks cited Robert A. Heinlein, Isaac Asimov, Arthur C. Clarke, Brian Aldiss, M. John Harrison and Dan Simmons as influences. The Crow Road, published in 1992, was adapted as a BBC television series. Banks continued to write both science fiction and mainstream. His final novel The Quarry appeared in June 2013, the month of his death.

Banks published work under two names. His parents had meant to name him "Iain Menzies Banks", but his father mistakenly registered him as "Iain Banks". Banks still used the middle name and submitted The Wasp Factory for publication as "Iain M. Banks". Banks's editor inquired about the possibility of omitting the 'M' as it appeared "too fussy" and the potential existed for confusion with Rosie M. Banks, a romantic novelist in the Jeeves novels by P. G. Wodehouse; Banks agreed to the omission. After three mainstream novels, Banks's publishers agreed to publish his first science fiction (SF) novel Consider Phlebas. To create a distinction between the mainstream and the SF, Banks suggested returning the 'M' to his name, which was then used in all of his science fiction works.

By his death in June 2013, Banks had published 26 novels. A 27th novel The Quarry was published posthumously. His final work, a poetry collection, appeared in February 2015. In an interview in January 2013, he also mentioned he had the plot idea for another novel in the Culture series, which would most likely have been his next book and was planned for publication in 2014. A project to publish Banks's unseen early drawings, maps and sketches from the Culture universe alongs with his writings and notes on the setting was underway in February 2018. In 2021, the delayed single volume of The Culture: Notes and Drawings was cancelled and replaced with two separate volumes: a landscape artbook of The Culture: The Drawings and a companion volume containing notes, excerpts and new text from Ken MacLeod. The Culture: The Drawings was released on 7 November 2023, while the still-untitled companion volume was scheduled for late 2024.

Banks wrote in various categories, but enjoyed science fiction most.

In September 2012 Banks became a Guest of Honour at the 2014 World Science Fiction Convention, Loncon 3.

Radio and television
Banks was the subject of The Strange Worlds of Iain Banks South Bank Show (1997), a TV documentary that examined his mainstream writing, and was an in-studio guest for the final episode of Marc Riley's Rocket Science radio show, broadcast on BBC Radio 6 Music. An audio version of The Business, set to contemporary music, arranged by Paul Oakenfold, was broadcast in October 1999 on Galaxy Fm as the tenth Urban Soundtracks. Banks's The State of the Art, adapted for radio by Paul Cornell, was broadcast on BBC Radio 4 in 2009 with Nadia Molinari producing and directing. In 1998 Espedair Street was dramatised as a serial for Radio 4, presented by Paul Gambaccini in the style of a Radio 1 documentary.

In 2011 Banks featured on the BBC Radio 4 programme Saturday Live. Banks reaffirmed his atheism in this appearance, explaining death as an important "part of the totality of life" that should be treated realistically instead of feared.

Banks appeared on the BBC television programme Question Time, a show that features political discussion. In 2006 he captained a team of writers to victory in a special series of BBC Two's University Challenge. Banks also won a 2006 edition of BBC One's Celebrity Mastermind; the author selected "Malt whisky and the distilleries of Scotland" as his specialist subject.

His final interview was with Kirsty Wark, broadcast on BBC2 Scotland as Iain Banks: Raw Spirit 12 June 2013.

BBC One Scotland and BBC2 broadcast an adaptation of his novel Stonemouth in June 2015.

Theatre
Banks was involved in the stage production The Curse of Iain Banks, written by Maxton Walker and performed at the Edinburgh Fringe festival in 1999. Banks collaborated frequently with its soundtrack composer Gary Lloyd, for instance on a song collection they co-composed as a tribute to the fictional band Frozen Gold from Banks's novel Espedair Street. Lloyd also scored for a spoken word and music production of his novel The Bridge, which Banks himself voiced and which featured a cast of 40 musicians, released on CD by Codex Records in 1996. Lloyd recorded Banks for including in the play as a disembodied voice of himself in one of the cast member's dreams. Lloyd explained his collaboration with Banks on their first versions of Espedair Street (later versions being dated between 2005 and 2013) in a Guardian article prior to the opening of The Curse of Iain Banks: When he [Banks] first played them to me, I think he was worried that they might not be up to scratch (some of them dated back to 1973 and had never been heard). He needn't have worried. They're fantastic. We're slaving away to get the songs to the stage where we can go into the studio and make a demo. Iain bashes out melodies on his state-of-the-art Apple Mac in Edinburgh and sends them down to me in Chester where I put them onto my Atari.

Politics
Banks's political stance has been termed "left of centre" and in 2002 he endorsed the Scottish Socialist Party.

He was an Honorary Associate of the National Secular Society and a Distinguished Supporter of the Humanist Society Scotland. As a signatory to the Declaration of Calton Hill, he supported Scottish independence. In November 2012, Banks backed the campaign group emerging from the Radical Independence Conference held in that month. He opined that the independence movement was marked by cooperation: "Scots just seem to be more communitarian than the consensus expressed by the UK population as a whole."

In late 2004, Banks joined a group of UK politicians and media figures campaigning to have Prime Minister Tony Blair impeached after the 2003 invasion of Iraq. In protest, he cut up his passport and posted it to 10 Downing Street. In a Socialist Review interview, Banks explained that his passport protest occurred after he had "abandoned the idea of crashing my Land Rover through the gates of Fife dockyard, after spotting the guys armed with machine guns." Banks relayed his concerns about the Iraq invasion in his book Raw Spirit and through the protagonist Alban McGill in the novel The Steep Approach to Garbadale, who confronts another character with arguments of a similar kind.

In 2010, Banks called for a cultural and educational boycott of Israel after the Gaza flotilla raid incident. In a letter to The Guardian newspaper, Banks said he had instructed his agent to turn down any further book translation deals with Israeli publishers: Appeals to reason, international law, U. N. resolutions and simple human decency mean – it is now obvious – nothing to Israel... I would urge all writers, artists and others in the creative arts, as well as those academics engaging in joint educational projects with Israeli institutions, to consider doing everything they can to convince Israel of its moral degradation and ethical isolation, preferably by simply having nothing more to do with this outlaw state.

An extract from Banks's contribution to the written collection Generation Palestine: Voices from the Boycott, Divestment and Sanctions Movement, entitled "Our People", appeared in The Guardian in the wake of the author's cancer revelation. The extract conveys the author's support for the Boycott, Divestment and Sanctions (BDS) campaign issued by a Palestinian civil society against Israel until the country complies with what it holds are international law and Palestinian rights. This commenced in 2005 and applies lessons from Banks's experience with South Africa's apartheid era. The continuation of Banks's boycott of Israeli publishers for the sale of rights to his novels was confirmed in the extract and Banks further explained, "I don't buy Israeli-sourced products or food, and my partner and I try to support Palestinian-sourced products wherever possible."

Personal life
Banks met his first wife Annie in London before the 1984 release of his first book. They lived in Faversham in the south of England, then split up in 1988. Banks returned to Edinburgh and dated another woman for two years. Iain and Annie were reconciled a year later and they moved to Fife. They were married in Hawaii in 1992, but in 2005, after 15 years of marriage, they separated.

In 1998 Banks was in a near-fatal accident when his car rolled off the road. In February 2007, Banks sold his extensive car collection, including a 3.2-litre Porsche Boxster, a Porsche 911 Turbo, a 3.8-litre Jaguar Mark II, a 5-litre BMW M5 and a daily-use diesel Land Rover Defender, whose power he had boosted by about 50 per cent. All these Banks exchanged for a Lexus RX 400h hybrid – later replaced by a diesel Toyota Yaris, and said in future he would fly only in emergencies.

In April 2012 Banks became the "Acting Honorary Non-Executive Figurehead President Elect pro tem (trainee)" of the Science Fiction Book Club based in London. The title was his creation and on 3 October 2012 Banks accepted a T-shirt inscribed with it.

From 2006 Banks lived in North Queensferry on the north side of the Firth of Forth, with his girlfriend Adele Hartley, an author and founder of the Dead by Dawn film festival. She and Banks had been friends since the early 1980s, and married on 29 March 2013 after he asked her to "do me the honour of becoming my widow."

Illness and death
On 3 April 2013, Banks announced on his website and on one set up by him and some friends that he had been diagnosed with terminal gallbladder cancer and was unlikely to live beyond a year. He stated he would be withdrawing from all public engagements and that The Quarry would be his last novel. The dates of publication of The Quarry were brought forward at Banks's request, to 20 June 2013 in the UK and 25 June 2013 in the US and Canada. He died on 9 June 2013.

Remembrance and tribute
Banks's publisher called him "an irreplaceable part of the literary world". This was reaffirmed by a fellow Scottish author and friend since secondary school Ken MacLeod: his death "left a large gap in the Scottish literary scene as well as the wider English-speaking world." British author Charles Stross wrote, "One of the giants of 20th and 21st century Scottish literature has left the building." Authors, including Neil Gaiman, Ian Rankin, Alastair Reynolds and David Brin also paid tribute in blogs and elsewhere.

The asteroid 5099 Iainbanks was named after him shortly after his death. On 23 January 2015, SpaceX's CEO Elon Musk named two of the firm's autonomous spaceport drone ships Just Read The Instructions and Of Course I Still Love You, after ships in Banks's novel The Player of Games. Another, A Shortfall of Gravitas, began construction in 2018. This refers to the ship Experiencing A Significant Gravitas Shortfall, first mentioned in Look to Windward.

The Red Virgin and the Vision of Utopia, the 2016 graphic biography of Louise Michel by Mary M. Talbot and Bryan Talbot, is "Dedicated to the memory of Iain (M) Banks, friend and sorely missed creator of socialist utopias."

Empire Games, the seventh book in The Merchant Princes series by Charles Stross published in 2017, is dedicated "For Iain M. Banks, who painted a picture of a better way."

On 13 May 2019, the Five Deeps Expedition broke the deepest ocean dive record in the DSV Limiting Factor. The support ship was named DSSV Pressure Drop. Both vessels were named after ships in the Culture series, which is much admired by the explorer Victor Vescovo, also the financial sponsor behind Limiting Factor's design and construction. They also have landers named "Flere," "Skaff," and "Closp," named after Culture drones.

Awards and nominations
Iain Banks received the following literary awards and nominations:

Non-SF works
Banks's non-SF work comprises fourteen novels and one non-fiction book. Many of his novels contain elements of autobiography, and feature various locations in his native Scotland. Raw Spirit (subtitled In Search of the Perfect Dram) is a travel book of Banks's visits to the distilleries of Scotland in search of the finest whisky, including his musings on other subjects such as cars and politics.

Fiction

 * The Wasp Factory (1984). London: Macmillan. ISBN 0-333-36380-9
 * Walking on Glass (1985). London: Macmillan. ISBN 0-333-37986-1
 * The Bridge (1986). London: Macmillan. ISBN 0-333-41285-0
 * Espedair Street (1987). London: Macmillan. ISBN 0-333-44916-9. Adapted for BBC radio in 1998 (directed by Dave Batchelor).
 * Canal Dreams (1989). London: Macmillan. ISBN 0-333-51768-7
 * The Crow Road (1992). London: Scribners. ISBN 0-356-20652-1. Adapted for BBC TV in 1996 (directed by Gavin Millar).
 * Complicity (1993). London: Little, Brown Book Group. ISBN 0-316-90688-3. Filmed in 2000 (directed by Gavin Millar); retitled Retribution for its US DVD/video release.
 * Whit (1995). London: Little, Brown Book Group. ISBN 0-316-91436-3
 * A Song of Stone (1997). London: Abacus. ISBN 0-316-64016-6
 * The Business (1999). London: Little, Brown Book Group. ISBN 0-316-64844-2.
 * Dead Air (2002). London: Little, Brown Book Group. ISBN 0-316-86054-9
 * The Steep Approach to Garbadale (2007). London: Little, Brown Book Group. ISBN 978-0-316-73105-8
 * Stonemouth (2012). London: Little, Brown Book Group. ISBN 978-1-4087-0250-5. Adapted for BBC TV for broadcast in 2015 (directed by Charles Martin.)
 * The Quarry (2013). London: Little, Brown Book Group. ISBN 978-1-4087-0394-6

Non-fiction

 * Raw Spirit (2003). London: Century. ISBN 1-84413-195-5 – a travelogue of Scotland and its whisky distilleries.
 * Poems (with Ken MacLeod) (2015). London: Little, Brown Group. ISBN 978-1-4087-0587-2

Science fiction
Banks wrote thirteen SF novels, nine of which were part of the Culture series, and a short story collection called The State of the Art (1991), which includes some stories set in the same universe. These works focus upon characters that are usually on the margins of the Culture, a post-scarcity anarchist utopia. In the same universe are other civilizations, which the Culture sometimes attempts to influence or "contact", occasionally resulting in conflict. The culture has achieved utopia by handing control of all of their worlds and ships over to sentient artificial intelligences referred to as "Minds".

The Culture novels

 * 1) Consider Phlebas (1987). London: Macmillan. ISBN 0-333-44138-9
 * 2) The Player of Games (1988). London: Macmillan. ISBN 0-333-47110-5
 * 3) The State of the Art (1989). London: Orbit. ISBN 0-356-19669-0 – also included below in short fiction collections, but included here because it is considered part of the Culture series.
 * 4) Use of Weapons (1990). London: Orbit. ISBN 0-356-19160-5
 * 5) Excession (1996). London: Orbit. ISBN 1-85723-394-8
 * 6) Inversions (1998). London: Orbit. ISBN 1-85723-626-2
 * 7) Look to Windward (2000). London: Orbit. ISBN 1-85723-969-5
 * 8) Matter (2008). London: Orbit. ISBN 978-1-84149-417-3
 * 9) Surface Detail (2010). London: Orbit. ISBN 978-1-84149-893-5
 * 10) The Hydrogen Sonata (2012). London: Orbit. ISBN 978-0-356-50150-5

The Culture reference books

 * The Culture: The Drawings (2023). London: Orbit. ISBN 978-0-356-51942-5

Other novels

 * Against a Dark Background (1993). London: Orbit. ISBN 1-85723-031-0
 * Feersum Endjinn (1994). London: Orbit. ISBN 1-85723-235-6
 * The Algebraist (2004). London: Orbit. ISBN 1-84149-155-1
 * Transition (2009). London: Little, Brown Book Group. ISBN 978-0-316-73107-2. (Published in the United States as Iain M. Banks.)

Short fiction collections

 * The State of the Art (1991). London: Orbit. ISBN 0-356-19669-0
 * Includes three short works set in the Culture universe. It also includes works of fiction more characteristic of Banks's writing published as Iain Banks. A radio version of the title story was transmitted by Radio 4 in 2009.
 * The Spheres (Birmingham Science Fiction Group, 2010)
 * Includes 'The Spheres', excised from the original draft of Transition; and 'The Secret Courtyard', excised from Matter. Limited edition of 500, to mark Novacon 40.

Introductions
Banks wrote introductions for works by other writers including:
 * Viriconium (1988) by M. John Harrison, the Unwin edition, ISBN 0-04-440245-7.
 * The Adventures of Luther Arkwright: Book 3, Götterdämmerung (1989) by Bryan Talbot from Proutt Publishing, ISBN 0-907865-03-8.
 * The Orbit Science Fiction Yearbook Three (1990) edited by David S. Garnett, ISBN 0-07-088833-7.
 * The Human Front (2001) by Ken MacLeod, the PS Publishing edition, ISBN 1-902880-30-7 (hbk) and ISBN 1-902880-31-5 (pbk).

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Iain Banks


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Talk:Iain Banks

Inversions as a Culture novel?
I'd like to dispute Inversions being a Culture novel. As far as I'm concerned, the vague assertion that the Culture may be involved in the storyline doesn't make it a Culture novel, any more than references to death make one of Shakespeare's comedies into a tragedy... Excession 22:20, 30 December 2005 (UTC)

 * To quote Banks, "Inversions was an attempt to write a Culture novel that wasn't." This is unlike Feersum Endjinn which Banks confirmed being entirely unrelated to Culture, after some people had interpreted it otherwise. Aapo Laitinen 12:39, 31 December 2005 (UTC)

Personally, I found "Inversions" to be one of the most pleasing Culture novels. The impact of the story was inherent in its subtlety. I'd recommend it to anyone as a non-genre but they'd have to read "Consider Phlebas" first! P.S: (user: "Exession") "...The vague assertion that the Culture may be involved in the storyline..." Way to miss the whole point... Twat! —Preceding unsigned comment added by Ruskin35 (talk • contribs) 00:51, 17 January 2009 (UTC)

Of course it's a Culture novel! If the 'vague' assertions are not enough for you, how about the knife missile? //roger.duprat.copenhagen — Preceding unsigned comment added by 94.191.176.245 (talk) 06:48, 30 November 2011 (UTC)

One of the novel's key strengths is actually not knowing that it is a Culture novel. Unlike every other Culture novel, Inversions does not feature "A Culture Novel" on the cover, and this is precisely because that fact is not revealed until the very end. It is probably the most genius and memorable plot twist I have ever come across and the "surprise" is entirely intentional by the author. For this reason it is only correct for it to be classified outside of the list of Culture novels, but not because it barely makes reference to the Culture, but because otherwise the whole plot twist is given away. 2A02:8084:9841:2900:9896:9AF0:4F6F:45A3 (talk) 19:49, 22 February 2019 (UTC)

Wikipedia doesn't care about spoilers. The book is listed as part of the Culture series on the author's website. That would seem to settle the issue. 64.141.80.200 (talk) 18:50, 22 March 2019 (UTC)

Transition, science fiction or not?
The book Transition has been moved from the Fiction section of the bibliography to the Science Fiction section. I think this is worth discussing. The edit comment cites evidence from the Wikipedia article about Transition, and also that it was nominated for a science fiction award. I don't think that is a particularly strong case as those are the opinions of others and the author himself described it as "a mainstream novel that borrows science fiction tropes" (see: http://news.bbc.co.uk/1/hi/sci/tech/7948058.stm). Personally I think the whole bibliography would be easier to read if it was arranged chronologically, then such demarcation issues would not arise, with entries along the lines of: 1984, The Wasp Factory, an Iain Banks novel. 1987, Consider Phlebas, an Iain M Banks Culture novel. --User:DaveH —Preceding undated comment added 12:45, 4 December 2017 (UTC)

 * Good idea. The attempt to separate his streams of work into two sections is far too subjective, and unnecessary. Those who claim that the nom de plume he used when writing one stream or the other is a definitive guide need to consider that in some markets some books were published under the other non de plume to the main markets. Greglocock (talk) 20:18, 4 December 2017 (UTC)

New TV series in development
Shouldn't this be mentioned? --Piotr Konieczny aka Prokonsul Piotrus| reply here 15:08, 13 August 2019 (UTC)

Ming and Aing
"He wrote mainstream fiction under the name Iain Banks and science fiction as Iain M. Banks"

Not this old cliché. The Bridge is a science fiction novel and is published under the "Iain Banks" label. As is Walking on Glass. — Preceding unsigned comment added by 213.205.241.45 (talk) 12:44, 25 October 2019 (UTC)

Nationality - Scottish or British?
I see a later editor removed the field entirely - workable, but a bit drastic IMHO. So anyway, last week I was checking my watchlist, and a sock had added "British" to the nationality field of the infobox, apparently the first wikipedia edit by that address. If you are at ALL familiar with the works and life of Iain M Banks like I am, you would know that he fervently disliked being called anything but Scottish. I noted this and reverted it. The next time I checked I found this in the edit history of the page: "90.243.22.131 Undid revision 1010729790 by Timmccloud (talk)] It's not a matter of identity or style. Banks' passport said 'British' and that is his legal and meaningful nationality" This was done by an anonymous sock, without any references for the changes made, and had begun an edit war with me. He's been warned about the concept of edit warring, and I told him to weigh in here, like he should. I still think this deserves a discussion, what is the consensus? Scottish or British? Whatever consensus is, it shall be done. VOTING:
 * British - I assume this is the IP sock vote 90.243.18.24
 * Scottish - This was Iain's personal preference, it's stated in all his book cover bios as such, and I believe it should be reflected as such in the info box. Timmccloud (talk) 00:05, 17 March 2021 (UTC)
 * Leave the field out - I'm the editor who removed the field. As I said in my edit note, it is an optional field, nothing is lost by not having it. Place of birth, place of death, education... there's plenty of content in the article that shows Banks is Scottish without needing to have a nationality line in the infobox. Looking at some other articles on Scottish writers as examples, Muriel Spark and William McIlvanney do not have a nationality field, while [James Kelman]], Ian Rankin and Robert Louis Stevenson have their nationalities listed as Scottish. I don't see the value in having a nationality field when it's already strongly implied; it seems to be most useful for highlighting when a writer is maybe not the nationality you'd expect. But if it must be there, it clearly should be Scottish - to counter the passport argument, that's what the Citizenship field is for. H. Carver (talk) 01:46, 17 March 2021 (UTC)
 * Comment . British With all due respect, British nationality is a legal concept. What about Scottish nationality? The term just redirects to the same place British one does. Internationally, Banks is considered a British writer. Maybe a footnote can be added with a comment that he preferred the other term? --Piotr Konieczny aka Prokonsul Piotrus| reply here  08:50, 17 March 2021 (UTC)
 * I do not believe errors and simplifications made internationally should be cited as precedent. Great Britain is an United Kingdom of four nations. The distinction exists. Perhaps we should change Drake's page because he's internationally known as a North American rapper? Perhaps we should go through the many Wikipedia articles of Catalan people and change them all to being Spanish? There's also the fact that while there is nationally as a legal concept, the documentation for the template does not say that is the definition applied for the template. H. Carver (talk) 12:24, 17 March 2021 (UTC)
 * Thanks, your comment made me change my view. Until Scotland becomes independent, there is no such thing as Scottish nationality. Although as I said, I'd be fine with a footnote about his personal views on this. --Piotr Konieczny aka Prokonsul Piotrus| reply here 03:49, 19 March 2021 (UTC)

If interested editors haven't already seen it, there exists an essay that discusses the issues of British nationality in some detail, and may help inform - Nationality of people from the United Kingdom H. Carver (talk) 12:25, 17 March 2021 (UTC)

British isn't a nationality. Banks was Scottish.

Northern Ireland is not in Great Britain
This is to answer the comment above that says that there are 4 countries in GB. 82.36.70.45 (talk) 04:14, 28 February 2023 (UTC)
 * Yes, the editor wrote "Great Britain is an United Kingdom of four nations". He's confusing two things. Great Britain is an island, consisting of Wales, Scotland and England. It is essentially a geographical, not a political, concept (although it was once a political entity when the Kingdom of Great Britain existed 1707-1800). Nowadays, it provides three of the four home nations of the United Kingdom of Great Britain and Northern Ireland. The other one, Northern Ireland, is a part of the United Kingdom, but is most definitely not part of Great Britain. --   Jack of Oz   [pleasantries]  06:30, 28 February 2023 (UTC)

title: Talk:Iain Banks


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Incunable

An incunable or incunabulum (plural incunables or incunabula, respectively) is a book, pamphlet, or broadside that was printed in the earliest stages of printing in Europe, up to the year 1500. Incunabula were produced before the printing press became widespread on the continent and are distinct from manuscripts, which are documents written by hand. Some authorities on the history of printing include block books from the same time period as incunabula, whereas others limit the term to works printed using movable type.

there are about 30,000 distinct incunable editions known. The probable number of surviving individual copies is much higher, estimated at 125,000 in Germany alone. Through statistical analysis, it is estimated that the number of lost editions is at least 20,000. Around 550,000 copies of around 27,500 different works have been preserved worldwide.

Terminology
Incunable is the anglicised form of incunabulum, reconstructed singular of Latin incunabula, which meant "swaddling clothes", or "cradle", which could metaphorically refer to "the earliest stages or first traces in the development". A former term for incunable is fifteener, meaning "fifteenth-century edition".

The term incunabula was first used in the context of printing by the Dutch physician and humanist Hadrianus Junius (Adriaen de Jonghe, 1511–1575), in a passage in his work Batavia (written in 1569; published posthumously in 1588). He referred to a period "inter prima artis [typographicae] incunabula" ("in the first infancy of the typographic art"). The term has sometimes been incorrectly attributed to Bernhard von Mallinckrodt (1591–1664), in his Latin pamphlet De ortu ac progressu artis typographicae ("On the rise and progress of the typographic art"; 1640), but he was quoting Junius.

The term incunabula came to denote printed books themselves in the late 17th century. It is not found in English before the mid-19th century.

Junius set an end-date of 1500 to his era of incunabula, which remains the convention in modern bibliographical scholarship. This convenient but arbitrary end-date for identifying a printed book as an incunable does not reflect changes in the printing process, and many books printed for some years after 1500 are visually indistinguishable from incunables. The term "post-incunable" is now used to refer to books printed after 1500 up to 1520 or 1540, without general agreement. From around this period the dating of any edition becomes easier, as the practice of printing the place and year of publication using a colophon or on the title page became more widespread.

Types
There are two types of printed incunabula: the block book, printed from a single carved or sculpted wooden block for each page (the same process as the woodcut in art, called xylographic); and the typographic book, made by individual cast-metal movable type pieces on a printing press. Many authors reserve the term "incunabula" for the latter.

The spread of printing to cities both in the North and in Italy ensured that there was great variety in the texts and the styles which appeared. Many early typefaces were modelled on local writing or derived from various European Gothic scripts, but there were also some derived from documentary scripts like Caxton's, and, particularly in Italy, types modelled on handwritten scripts and calligraphy used by humanists.

Printers congregated in urban centres where there were scholars, ecclesiastics, lawyers, and nobles and professionals who formed their major customer base. Standard works in Latin inherited from the medieval tradition formed the bulk of the earliest printed works, but as books became cheaper, vernacular works (or translations into vernaculars of standard works) began to appear.

Famous examples

Famous incunabula include two from Mainz, the Gutenberg Bible of 1455 and the Peregrinatio in terram sanctam of 1486, printed and illustrated by Erhard Reuwich; the Nuremberg Chronicle written by Hartmann Schedel and printed by Anton Koberger in 1493; and the Hypnerotomachia Poliphili printed by Aldus Manutius with important illustrations by an unknown artist.

Other printers of incunabula were Günther Zainer of Augsburg, Johannes Mentelin and Heinrich Eggestein of Strasbourg, Heinrich Gran of Haguenau, Johann Amerbach of Basel, William Caxton of Bruges and London, and Nicolas Jenson of Venice. The first incunable to have woodcut illustrations was Ulrich Boner's Der Edelstein, printed by Albrecht Pfister in Bamberg in 1461.

A finding in 2015 brought evidence of quires, as claimed by extensive research, printed in 1444–1446, possibly assigned to Procopius Waldvogel in Avignon, France.

Post-incunable
Many incunabula are undated, needing complex bibliographical analysis to place them correctly. The post-incunabula period marks a time of development during which the printed book evolved fully as a mature artefact with a standard format. After about 1540 books tended to conform to a template that included the author, title-page, date, seller, and place of printing. This makes it much easier to identify any particular edition.

As noted above, the end date for identifying a printed book as an incunable is convenient but was chosen arbitrarily; it does not reflect any notable developments in the printing process around the year 1500. Books printed for a number of years after 1500 continued to look much like incunables, with the notable exception of the small format books printed in italic type introduced by Aldus Manutius in 1501. The term post-incunable is sometimes used to refer to books printed "after 1500—how long after, the experts have not yet agreed." For books printed in the UK, the term generally covers 1501–1520, and for books printed in mainland Europe, 1501–1540.

Statistical data

The data in this section were derived from the Incunabula Short-Title Catalogue (ISTC).

The number of printing towns and cities stands at 282. These are situated in some 18 countries in terms of present-day boundaries. In descending order of the number of editions printed in each, these are: Italy, Germany, France, Netherlands, Switzerland, Spain, Belgium, England, Austria, the Czech Republic, Portugal, Poland, Sweden, Denmark, Turkey, Croatia, Serbia, Montenegro, and Hungary (see diagram).

The following table shows the 20 main 15th century printing locations; as with all data in this section, exact figures are given, but should be treated as close estimates (the total editions recorded in ISTC at August 2016 is 30,518): The 18 languages that incunabula are printed in, in descending order, are: Latin, German, Italian, French, Dutch, Spanish, English, Hebrew, Catalan, Czech, Greek, Church Slavonic, Portuguese, Swedish, Breton, Danish, Frisian and Sardinian (see diagram).

Only about one edition in ten (i.e. just over 3,000) has any illustrations, woodcuts or metalcuts.

The "commonest" incunable is Schedel's Nuremberg Chronicle ("Liber Chronicarum") of 1493, with about 1,250 surviving copies (which is also the most heavily illustrated). Many incunabula are unique, but on average about 18 copies survive of each. This makes the Gutenberg Bible, at 48 or 49 known copies, a relatively common (though extremely valuable) edition. Counting extant incunabula is complicated by the fact that most libraries consider a single volume of a multi-volume work as a separate item, as well as fragments or copies lacking more than half the total leaves. A complete incunable may consist of a slip, or up to ten volumes.

In terms of format, the 30,000-odd editions comprise: 2,000 broadsides, 9,000 folios, 15,000 quartos, 3,000 octavos, 18 12mos, 230 16mos, 20 32mos, and 3 64mos.

ISTC at present cites 528 extant copies of books printed by Caxton, which together with 128 fragments makes 656 in total, though many are broadsides or very imperfect (incomplete).

Apart from migration to mainly North American and Japanese universities, there has been little movement of incunabula in the last five centuries. None were printed in the Southern Hemisphere, and the latter appears to possess less than 2,000 copies, about 97.75% remain north of the equator. However, many incunabula are sold at auction or through the rare book trade every year.

Major collections
The British Library's Incunabula Short Title Catalogue now records over 29,000 titles, of which around 27,400 are incunabula editions (not all unique works). Studies of incunabula began in the 17th century. Michel Maittaire (1667–1747) and Georg Wolfgang Panzer (1729–1805) arranged printed material chronologically in annals format, and in the first half of the 19th century, Ludwig Hain published the Repertorium bibliographicum—a checklist of incunabula arranged alphabetically by author: "Hain numbers" are still a reference point. Hain was expanded in subsequent editions, by Walter A. Copinger and Dietrich Reichling, but it is being superseded by the authoritative modern listing, a German catalogue, the Gesamtkatalog der Wiegendrucke, which has been under way since 1925 and is still being compiled at the Staatsbibliothek zu Berlin. North American holdings were listed by Frederick R. Goff and a worldwide union catalogue is provided by the Incunabula Short Title Catalogue.

Notable collections with more than 1,000 incunabula include:

title: Incunable


text:
Talk:Islamic calendar

an error which is causing harm to sentiments of billions muslims.
I know this not a delibrate action and the intentions are not ill but please take down the illustration where it says to be an illustration of our Prophet Muhammad PBUH because this will unintentionally cause billions of Muslim's emotions being hurt as in our belief associating portraits of our Holy Prophet PBUH is disrespect. 139.135.36.84 (talk) 15:25, 19 March 2023 (UTC)

 * Per long-established consensus and as explained in the headers at the top of this page, Wikipedia is not censored. —C.Fred (talk) 16:47, 19 March 2023 (UTC)
 * For the long explanation, please read the "frequently asked questions" page that explains how Wikipedia handles this question. --𝕁𝕄𝔽 (talk) 16:53, 19 March 2023 (UTC)
 * And to understand the theological and historical basis (or not) for the belief, see the article "Depictions of Muhammad". --𝕁𝕄𝔽 (talk) 18:28, 19 March 2023 (UTC)

Phase period vs Gregorian calendar
I've edited the Months section to replace a direct quote from the Astronomical Almanac supplement ("As a result, the cycle of twelve lunar months regresses through the seasons over a period of about 33 [solar] years") with a less ambiguous explanation ("This causes the Hijri calendar to gain a year against the Gregorian calendar over a period of 33 solar years, though dates themselves do not correspond from cycle to cycle"). This clarifies that the source is referring to the phase difference of the two calendars, not the period after which a given date in one will land on the same date in the other.

I unfortunately have not been able to find a source for what that period is, mostly because I'm not familiar enough with horology to know the best term to use to find one, but hopefully the fact that I'm clearing up a potential misinterpretation of the provided source (and one that can be seen repeated elsewhere on the internet, despite how implausible it becomes once one stops to think about it for more than a moment) means a new source isn't strictly necessary and my edit can stand until someone else can clean up behind me.

OR, but a fun fact all the same: based on the (tabular) Islamic calendar averaging 354.37 days per year and the Georgian averaging 365.2425, period after which the pattern of corresponding dates between the two calendars will repeat would be indicated by the cumulative days' divergence being divisible by the average number of days. I ran out of rows in Excel trying to calculate that, which means it's not below 1,048,577 years. I'm sure there's a way to calculate it, as Abramsky did for the Hebrew calendar's 14.39 billion years - but I'm not a mathematician either, so I'll leave that to some other poor sod!

Ded10c Talk · Contrib's 16:10, 7 April 2023 (UTC)
 * I'm not convinced that you have come up with the right solution here, so I am reverting for now, per WP:BRD. I appreciate the clarity problem but the AAS is more NPOV: it expresses the movement strictly with respect to the seasons and makes no reference to a calendar that may be seen as Christian. (Nitpickingly so, IMO, but technically correct.) Also, I don't see the ambiguity? the AAS doesn't mention dates at all. Can you think of a better way to express it? (I can't!) Meanwhile, rather than a full-on revert, I'll make your text a footnote. --𝕁𝕄𝔽 (talk) 20:14, 7 April 2023 (UTC)

Islam calendar, the year is 1445, and documented as it began

Friday, 16 July 622 C.E

622 AD, however the Persian Calendar below actually began in 622AD.

We did the math:

Persian is near to Islam and todays date (March 30, 2024 Gregorian) is the year 1402.

1402 + 622AD = the year 2024! 99.85.74.41 (talk) 08:26, 30 March 2024 (UTC)

Calendar image
Does anybody know with confidence whether this image is of a solar or a lunar Hijri calendar? 𝕁𝕄𝔽 (talk) 07:26, 19 April 2024 (UTC)

title: Talk:Islamic calendar


text:
Isotropy

In physics and geometry, isotropy is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence anisotropy. Anisotropy is also used to describe situations where properties vary systematically, dependent on direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented.

Mathematics
Within mathematics, isotropy has a few different meanings:
 * Isotropic manifolds: A manifold is isotropic if the geometry on the manifold is the same regardless of direction. A similar concept is homogeneity.
 * Isotropic quadratic form: A quadratic form q is said to be isotropic if there is a non-zero vector v such that q(v) = 0; such a v is an isotropic vector or null vector. In complex geometry, a line through the origin in the direction of an isotropic vector is an isotropic line.
 * Isotropic coordinates: Isotropic coordinates are coordinates on an isotropic chart for Lorentzian manifolds.
 * Isotropy group:An isotropy group is the group of isomorphisms from any object to itself in a groupoid. An isotropy representation is a representation of an isotropy group.
 * Isotropic position: A probability distribution over a vector space is in isotropic position if its covariance matrix is the identity.
 * Isotropic vector field: The vector field generated by a point source is said to be isotropic if, for any spherical neighborhood centered at the point source, the magnitude of the vector determined by any point on the sphere is invariant under a change in direction. For an example, starlight appears to be isotropic.

Physics

 * Quantum mechanics or particle physics: When a spinless particle (or even an unpolarized particle with spin) decays, the resulting decay distribution must be isotropic in the rest frame of the decaying particle - regardless of the detailed physics of the decay. This follows from rotational invariance of the Hamiltonian, which in turn is guaranteed for a spherically symmetric potential.

 * Gases: The kinetic theory of gases also exemplifies isotropy. It is assumed that the molecules move in random directions and as a consequence, there is an equal probability of a molecule moving in any direction. Thus when there are many molecules in the gas, with high probability there will be very similar numbers moving in one direction as any other, demonstrating approximate isotropy.

 * Fluid dynamics: Fluid flow is isotropic if there is no directional preference (e.g. in fully developed 3D turbulence). An example of anisotropy is in flows with a background density as gravity works in only one direction. The apparent surface separating two differing isotropic fluids would be referred to as an isotrope.

 * Thermal expansion: A solid is said to be isotropic if the expansion of solid is equal in all directions when thermal energy is provided to the solid.

 * Electromagnetics: An isotropic medium is one such that the permittivity, ε, and permeability, μ, of the medium are uniform in all directions of the medium, the simplest instance being free space.

 * Optics: Optical isotropy means having the same optical properties in all directions. The individual reflectance or transmittance of the domains is averaged for micro-heterogeneous samples if the macroscopic reflectance or transmittance is to be calculated. This can be verified simply by investigating, for example, a polycrystalline material under a polarizing microscope having the polarizers crossed: If the crystallites are larger than the resolution limit, they will be visible.

 * Cosmology: The cosmological principle, which underpins much of modern cosmology (including the Big Bang theory of the evolution of the observable universe), assumes that the universe is both isotropic and homogeneous, meaning that the universe has no preferred location (is the same everywhere) and has no preferred direction. Observations made in 2006 suggest that, on distance-scales much larger than  galaxies, galaxy clusters are "Great" features, but small compared to so-called multiverse scenarios.

Materials science
In the study of mechanical properties of materials, "isotropic" means having identical values of a property in all directions. This definition is also used in geology and mineralogy. Glass and metals are examples of isotropic materials. Common anisotropic materials include wood (because its material properties are different parallel to and perpendicular to the grain) and layered rocks such as slate.

Isotropic materials are useful since they are easier to shape, and their behavior is easier to predict. Anisotropic materials can be tailored to the forces an object is expected to experience. For example, the fibers in carbon fiber materials and rebars in reinforced concrete are oriented to withstand tension.

Microfabrication
In industrial processes, such as etching steps, "isotropic" means that the process proceeds at the same rate, regardless of direction. Simple chemical reaction and removal of a substrate by an acid, a solvent or a reactive gas is often very close to isotropic. Conversely, "anisotropic" means that the attack rate of the substrate is higher in a certain direction. Anisotropic etch processes, where vertical etch-rate is high but lateral etch-rate is very small, are essential processes in microfabrication of integrated circuits and MEMS devices.

Antenna (radio)
An isotropic antenna is an idealized "radiating element" used as a reference; an antenna that broadcasts power equally (calculated by the Poynting vector) in all directions. The gain of an arbitrary antenna is usually reported in decibels relative to an isotropic antenna, and is expressed as dBi or dB(i).

In cells (a.k.a. muscle fibers), the term "isotropic" refers to the light bands (I bands) that contribute to the striated pattern of the cells.

Pharmacology
While it is well established that the skin provides an ideal site for the administration of local and systemic drugs, it presents a formidable barrier to the permeation of most substances. Recently, isotropic formulations have been used extensively in dermatology for drug delivery.

Computer science

 * Imaging:A volume such as a computed tomography is said to have isotropic voxel spacing when the space between any two adjacent voxels is the same along each axis x, y, z. E.g., voxel spacing is isotropic if the center of voxel (i, j, k) is 1.38 mm from that of (i+1, j, k), 1.38 mm from that of (i, j+1, k) and 1.38 mm from that of (i, j, k+1) for all indices i, j, k.

Other sciences

 * Economics and geography: An isotropic region is a region that has the same properties everywhere. Such a region is a construction needed in many types of models.

title: Isotropy


text:
Talk:Instrument/Archive 1

Please see the entry under Musical Instruments before to much gets duplicated.

It is not right to use the title Instruments for just musical instruments. What about lab instruments, medical instruments etc.?

OK. Better? --WillWare

title: Talk:Instrument/Archive 1


text:
International Mathematical Union

The International Mathematical Union (IMU) is an international organization devoted to international cooperation in the field of mathematics across the world. It is a member of the International Science Council (ISC) and supports the International Congress of Mathematicians (ICM). Its members are national mathematics organizations from more than 80 countries.

The objectives of the International Mathematical Union are: promoting international cooperation in mathematics, supporting and assisting the International Congress of Mathematicians and other international scientific meetings/conferences, acknowledging outstanding research contributions to mathematics through the awarding of scientific prizes, and encouraging and supporting other international mathematical activities, considered likely to contribute to the development of mathematical science in any of its aspects, whether pure, applied, or educational.

History
The IMU was established in 1920, but dissolved in September 1932 and reestablished in 1950 at the Constitutive Convention in New York, de jure on September 10, 1951, when ten countries had become members. The last milestone was the General Assembly in March 1952, in Rome, Italy where the activities of the new IMU were inaugurated and the first Executive Committee, President and various commissions were elected. In 1952 the IMU was also readmitted to the ICSU. The past president of the Union is Carlos Kenig (2019–2022). The current president is Hiraku Nakajima.

At the 16th meeting of the IMU General Assembly in Bangalore, India, in August 2010, Berlin was chosen as the location of the permanent office of the IMU, which was opened on January 1, 2011, and is hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS), an institute of the Gottfried Wilhelm Leibniz Scientific Community, with about 120 scientists engaging in mathematical research applied to complex problems in industry and commerce.

Commissions and committees
IMU has a close relationship to mathematics education through its International Commission on Mathematical Instruction (ICMI). This commission is organized similarly to IMU with its own Executive Committee and General Assembly.

Developing countries are a high priority for the IMU and a significant percentage of its budget, including grants received from individuals, mathematical societies, foundations, and funding agencies, is spent on activities for developing countries. Since 2011 this has been coordinated by the Commission for Developing Countries (CDC).

The Committee for Women in Mathematics (CWM) is concerned with issues related to women in mathematics worldwide. It organizes the World Meeting for Women in Mathematics $((\mathrm{WM})^2)$ as a satellite event of ICM.

The International Commission on the History of Mathematics (ICHM) is operated jointly by the IMU and the Division of the History of Science (DHS) of the International Union of History and Philosophy of Science (IUHPS).

The Committee on Electronic Information and Communication (CEIC) advises IMU on matters concerning mathematical information, communication, and publishing.

Prizes
The scientific prizes awarded by the IMU, in the quadrennial International Congress of Mathematicians (ICM), are deemed to be some of the highest distinctions in the mathematical world. These are:
 * the Fields Medals (two to four awarded per Congress, since 1936);
 * the IMU Abacus Medal (previously known as the Rolf Nevanlinna Prize; awarded since 1986);
 * the Carl Friedrich Gauss Prize (since 2006);
 * the Chern Medal (since 2010); and
 * the Leelavati Award (since 2010).

Membership and General Assembly
The IMU's members are Member Countries and each Member country is represented through an Adhering Organization, which may be its principal academy, a mathematical society, its research council or some other institution or association of institutions, or an appropriate agency of its government. A country starting to develop its mathematical culture and interested in building links with mathematicians all over the world is invited to join IMU as an Associate Member. For the purpose of facilitating jointly sponsored activities and jointly pursuing the objectives of the IMU, multinational mathematical societies and professional societies can join IMU as an Affiliate Member. Every four years, the IMU membership gathers in a General Assembly (GA), which consists of delegates appointed by the Adhering Organizations, together with the members of the executive committee. All important decisions are made at the GA, including the election of the officers, establishment of commissions, the approval of the budget, and any changes to the statutes and by-laws.

Members and Associate Members
The IMU has 83 (full) Member countries and two Associate Members (Bangladesh and Paraguay, marked below by light grey background).

Affiliate members
The IMU has five affiliate members:

 * African Mathematical Union (AMU)
 * European Mathematical Society (EMS)
 * Mathematical Council of the Americas (MCofA)
 * Southeast Asian Mathematical Society (SEAMS)
 * Unión Matemática de América Latina y el Caribe (UMALCA)

Organization and Executive Committee
The International Mathematical Union is administered by an executive committee (EC) which conducts the business of the Union. The EC consists of the President, two vice-presidents, the Secretary, six Members-at-Large, all elected for a term of four years, and the Past President. The EC is responsible for all policy matters and for tasks, such as choosing the members of the ICM Program Committee and various prize committees.

Publications
Every two months IMU publishes an electronic newsletter, IMU-Net, that aims to improve communication between IMU and the worldwide mathematical community by reporting on decisions and recommendations of the Union, major international mathematical events and developments, and on other topics of general mathematical interest. IMU Bulletins are published annually with the aim to inform IMU's members about the Union's current activities. In 2009 IMU published the document Best Current Practices for Journals.

IMU’s Involvement in developing countries
The IMU took its first organized steps towards the promotion of mathematics in developing countries in the early 1970s and has, since then supported various activities. In 2010 IMU formed the Commission for Developing Countries (CDC) which brings together all of the past and current initiatives in support of mathematics and mathematicians in the developing world.

Some IMU Supported Initiatives:
 * Grants Program for Mathematicians: The Commission for Developing Countries supports research travel of mathematicians based in developing countries as well as mathematics research conferences in the developing world through its Grants Program which is open to mathematicians throughout the developing world, including countries that are not (yet) members of the IMU.
 * African Mathematics Millennium Science Initiative (AMMSI) is a network of mathematics centers in sub-Saharan Africa that organizes conferences and workshops, visiting lectureships and an extensive scholarship program for mathematics graduate students doing PhD work on the African continent.
 * Mentoring African Research in Mathematics (MARM): IMU supported the London Mathematical Society (LMS) in founding the MARM programme, which supports mathematics and its teaching in the countries of sub-Saharan Africa via a mentoring partnership between mathematicians in the United Kingdom and African colleagues, together with their students. It focuses on cultivating long-term mentoring relations between individual mathematicians and students.
 * Volunteer Lecturer Program (VLP) of IMU identifies mathematicians interested in contributing to the formation of young mathematicians in the developing world. The Volunteer Lecturer Program maintains a database of mathematic volunteers willing to offer month-long intensive courses at the advanced undergraduate or graduate level in degree programmes at universities in the developing world. IMU also seeks applications from universities and mathematics degree programmes in the developing world that are in need of volunteer lecturers, and that can provide the necessary conditions for productive collaboration in the teaching of advanced mathematics.

IMU also supports the International Commission on Mathematical Instruction (ICMI) with its programmes, exhibits and workshops in emerging countries, especially in Asia and Africa.

IMU released a report in 2008, Mathematics in Africa: Challenges and Opportunities, on the current state of mathematics in Africa and on opportunities for new initiatives to support mathematical development. In 2014, the IMU's Commission for Developing Countries CDC released an update of the report.

Additionally, reports about Mathematics in Latin America and the Caribbean and South East Asia. were published.

In July 2014 IMU released the report: The International Mathematical Union in the Developing World: Past, Present and Future (July 2014).

MENAO Symposium at the ICM
In 2014, the IMU held a day-long symposium prior to the opening of the International Congress of Mathematicians (ICM), entitled Mathematics in Emerging Nations: Achievements and Opportunities (MENAO). Approximately 260 participants from around the world, including representatives of embassies, scientific institutions, private business and foundations attended this session. Attendees heard inspiring stories of individual mathematicians and specific developing nations.

Presidents
List of presidents of the International Mathematical Union from 1952 to the present:

1952–1954: 🇺🇸 Marshall Harvey Stone (vice: 🇫🇷 Émile Borel, 🇩🇪 Erich Kamke)

1955–1958: 🇩🇪 Heinz Hopf (vice: 🇫🇷 Arnaud Denjoy, 🇬🇧 W. V. D. Hodge)

1959–1962: 🇫🇮 Rolf Nevanlinna (vice: Pavel Alexandrov, 🇺🇸 Marston Morse)

1963–1966: 🇨🇭 Georges de Rham (vice: 🇫🇷 Henri Cartan, 🇵🇱 Kazimierz Kuratowski)

1967–1970: 🇫🇷 Henri Cartan (vice: Mikhail Lavrentyev, 🇺🇸 Deane Montgomery)

1971–1974: 🇮🇳 K. S. Chandrasekharan (vice: 🇺🇸 Abraham Adrian Albert, Lev Pontryagin)

1975–1978: 🇺🇸 Deane Montgomery (vice: 🇬🇧 J. W. S. Cassels, 🇷🇴 Miron Nicolescu, 🇷🇴 Gheorghe Vrânceanu)

1979–1982: 🇸🇪 Lennart Carleson (vice: 🇯🇵 Masayoshi Nagata, Yuri Vasilyevich Prokhorov)

1983–1986: 🇩🇪 Jürgen Moser (vice: Ludvig Faddeev, 🇫🇷 Jean-Pierre Serre)

1987–1990: Ludvig Faddeev (vice: 🇦🇹 Walter Feit, 🇸🇪 Lars Hörmander)

1991–1994: 🇫🇷 Jacques-Louis Lions (vice: 🇬🇧 John H. Coates, 🇺🇸 David Mumford)

1995–1998: 🇺🇸 David Mumford (vice: 🇷🇺 Vladimir Arnold, 🇩🇪 Albrecht Dold)

1999–2002: 🇧🇷 Jacob Palis (vice: 🇬🇧 Simon Donaldson, 🇯🇵 Shigefumi Mori)

2003–2006: 🇬🇧 John M. Ball (vice: 🇫🇷 Jean-Michel Bismut, 🇯🇵 Masaki Kashiwara)

2007–2010: 🇭🇺 László Lovász (vice: 🇨🇳 Zhi-Ming Ma, 🇮🇹 Claudio Procesi)

2011–2014: 🇧🇪 Ingrid Daubechies (vice: 🇫🇷 Christiane Rousseau, 🇧🇷 Marcelo Viana)

2015–2018: 🇯🇵 Shigefumi Mori (vice: 🇦🇷 Alicia Dickenstein, 🇳🇿 Vaughan Jones)

2019–2022: 🇦🇷 Carlos Kenig (vice: 🇦🇺 Nalini Joshi, 🇿🇦 Loyiso Nongxa)

2023–2026: 🇯🇵Hiraku Nakajima (vice: 🇬🇧 Ulrike Tillmann, 🇨🇴 Tatiana Toro)

title: International Mathematical Union


text:
International Council for Science

The International Council for Science (ICSU, after its former name, International Council of Scientific Unions) was an international non-governmental organization devoted to international cooperation in the advancement of science. Its members were national scientific bodies and international scientific unions.

In 2017, the ICSU comprised 122 multi-disciplinary National Scientific Members, Associates and Observers representing 142 countries and 31 international, disciplinary Scientific Unions. ICSU also had 22 Scientific Associates. In July 2018, ICSU merged with the International Social Science Council (ISSC) to form the International Science Council (ISC) at a constituent general assembly in Paris.

Mission and principles
The ICSU's mission was to strengthen international science for the benefit of society. To do this, the ICSU mobilized the knowledge and resources of the international scientific community to:

 * Identify and address major issues of importance to science and society.
 * Facilitate interaction amongst scientists across all disciplines and from all countries.
 * Promote the participation of all scientists – regardless of race, citizenship, language, political stance, or gender – in the international scientific endeavour.
 * Provide independent, authoritative advice to stimulate constructive dialogue between the scientific community and governments, civil society, and the private sector."

Activities focused on three areas: International Research Collaboration, Science for Policy, and Universality of Science.

History
In July 2018, the ICSU became the International Science Council (ISC).

The ICSU itself was one of the oldest non-governmental organizations in the world, representing the evolution and expansion of two earlier bodies known as the International Association of Academies (IAA; 1899–1914) and the International Research Council (IRC; 1919–1931). In 1998, Members agreed that the Council's current composition and activities would be better reflected by modifying the name from the International Council of Scientific Unions to the International Council for Science, while its rich history and strong identity would be well served by retaining the existing acronym, ICSU.

Universality of science
"The Principle of Freedom and Responsibility in Science: the free and responsible practice of science is fundamental to scientific advancement and human and environmental well-being. Such practice, in all its aspects, requires freedom of movement, association, expression and communication for scientists, as well as equitable access to data, information, and other resources for research. It requires responsibility at all levels to carry out and communicate scientific work with integrity, respect, fairness, trustworthiness, and transparency, recognizing its benefits and possible harms. In advocating the free and responsible practice of science, the council promotes equitable opportunities for access to science and its benefits, and opposes discrimination based on such factors as ethnic origin, religion, citizenship, language, political or other opinion, sex, gender identity, sexual orientation, disability, or age."

The International Science Council's Committee on Freedom and Responsibility in Science (CFRS) "oversees this commitment and is the guardian of this work."

Structure
The ICSU Secretariat (20 staff in 2012) in Paris ensured the day-to-day planning and operations under the guidance of an elected executive board. Three Policy Committees − Committee on Scientific Planning and Review (CSPR), Committee on Freedom and Responsibility in the conduct of Science (CFRS) and Committee on Finance − assisted the executive board in its work and a General Assembly of all Members was convened every three years. ICSU has three Regional Offices − Africa, Asia and the Pacific as well as Latin America and the Caribbean.

Finances
The principal source of ICSU's finances was the contributions it receives from its members. Other sources of income are the framework contracts from UNESCO (United Nations Educational, Scientific and Cultural Organization) and grants and contracts from United Nations bodies, foundations and agencies, which are used to support the scientific activities of the ICSU Unions and interdisciplinary bodies.

title: International Council for Science