text:
February 13

Pre-1600

 * 962 – Emperor Otto I and Pope John XII co-sign the Diploma Ottonianum, recognizing John as ruler of Rome.
 * 1258 – Siege of Baghdad: Hulegu Khan, a prince of the Mongol Empire, orders his army to sack and plunder the city of Baghdad, which they had just captured.
 * 1322 – The central tower of Ely Cathedral falls on the night of 12th–13th.
 * 1462 – The Treaty of Westminster is finalised between Edward IV of England and the Scottish Lord of the Isles.
 * 1503 – Challenge of Barletta: Tournament between 13 Italian and 13 French knights near Barletta.
 * 1542 – Catherine Howard, the fifth wife of Henry VIII of England, is executed for adultery.

1601–1900

 * 1633 – Galileo Galilei arrives in Rome for his trial before the Inquisition.
 * 1642 – The Clergy Act becomes law, excluding bishops of the Church of England from serving in the House of Lords.
 * 1660 – With the accession of young Charles XI of Sweden, his regents begin negotiations to end the Second Northern War.
 * 1689 – William and Mary are proclaimed co-rulers of England.
 * 1692 – Massacre of Glencoe: Almost 80 Macdonalds at Glen Coe, Scotland are killed early in the morning for not promptly pledging allegiance to the new king, William of Orange.
 * 1726 – Parliament of Negrete between Mapuche and Spanish authorities in Chile bring an end to the Mapuche uprising of 1723–26.
 * 1755 – Treaty of Giyanti signed by VOC, Pakubuwono III and Prince Mangkubumi. The treaty divides the Javanese kingdom of Mataram into two: Sunanate of Surakarta and Sultanate of Yogyakarta.
 * 1849 – The delegation headed by Metropolitan bishop Andrei Șaguna hands out to the Emperor Franz Joseph I of Austria the General Petition of Romanian leaders in Transylvania, Banat and Bukovina, which demands that the Romanian nation be recognized.
 * 1861 – Italian unification: The Siege of Gaeta ends with the capitulation of the defending fortress, effectively bringing an end of the Kingdom of the Two Sicilies.
 * 1867 – Work begins on the covering of the Senne, burying Brussels's primary river and creating the modern central boulevards.
 * 1880 – Thomas Edison observes Thermionic emission.

1901–present

 * 1913 – The 13th Dalai Lama proclaims Tibetan independence following a period of domination by Manchu Qing dynasty and initiated a period of almost four decades of independence.
 * 1914 – Copyright: In New York City the American Society of Composers, Authors and Publishers is established to protect the copyrighted musical compositions of its members.
 * 1920 – The Negro National League is formed.
 * 1931 – The British Raj completes its transfer from Calcutta to New Delhi.
 * 1935 – A jury in Flemington, New Jersey finds Bruno Hauptmann guilty of the 1932 kidnapping and murder of the Lindbergh baby, the son of Charles Lindbergh.
 * 1945 – World War II: The siege of Budapest concludes with the unconditional surrender of German and Hungarian forces to the Red Army.
 * 1945  – World War II: Royal Air Force bombers are dispatched to Dresden, Germany to attack the city with a massive aerial bombardment.
 * 1951 – Korean War: Battle of Chipyong-ni, which represented the "high-water mark" of the Chinese incursion into South Korea, commences.
 * 1954 – Frank Selvy becomes the only NCAA Division I basketball player ever to score 100 points in a single game.
 * 1955 – Israel obtains four of the seven Dead Sea Scrolls.
 * 1955  – Twenty-nine people are killed when Sabena Flight 503 crashes into Monte Terminillo near Rieti, Italy.
 * 1960 – With the success of a nuclear test codenamed "Gerboise Bleue", France becomes the fourth country to possess nuclear weapons.
 * 1960  – Black college students stage the first of the Nashville sit-ins at three lunch counters in Nashville, Tennessee.
 * 1961 – An allegedly 500,000-year-old rock is discovered near Olancha, California, US, that appears to anachronistically encase a spark plug.
 * 1967 – American researchers discover the Madrid Codices by Leonardo da Vinci in the National Library of Spain.
 * 1975 – Fire at One World Trade Center (North Tower) of the World Trade Center in New York.
 * 1978 – Hilton bombing: A bomb explodes in a refuse truck outside the Hilton Hotel in Sydney, Australia, killing two refuse collectors and a policeman.
 * 1979 – An intense windstorm strikes western Washington and sinks a 0.5 mi long section of the Hood Canal Bridge.
 * 1981 – A series of sewer explosions destroys more than two miles of streets in Louisville, Kentucky.
 * 1983 – A cinema fire in Turin, Italy, kills 64 people.
 * 1984 – Konstantin Chernenko succeeds the late Yuri Andropov as general secretary of the Communist Party of the Soviet Union.
 * 1990 – German reunification: An agreement is reached on a two-stage plan to reunite Germany.
 * 1991 – Gulf War: Two laser-guided "smart bombs" destroy the Amiriyah shelter in Baghdad. Allied forces said the bunker was being used as a military communications outpost, but over 400 Iraqi civilians inside were killed.
 * 1996 – The Nepalese Civil War is initiated in the Kingdom of Nepal by the Communist Party of Nepal (Maoist-Centre).
 * 2001 – An earthquake measuring 7.6 on the Richter magnitude scale hits El Salvador, killing at least 944.
 * 2004 – The Harvard–Smithsonian Center for Astrophysics announces the discovery of the universe's largest known diamond, white dwarf star BPM 37093. Astronomers named this star "Lucy" after The Beatles' song "Lucy in the Sky with Diamonds".
 * 2007 – Taiwan opposition leader Ma Ying-jeou resigns as the chairman of the Kuomintang party after being indicted on charges of embezzlement during his tenure as the mayor of Taipei; Ma also announces his candidacy for the 2008 presidential election.
 * 2008 – Australian Prime Minister Kevin Rudd makes a historic apology to the Indigenous Australians and the Stolen Generations.
 * 2010 – A bomb explodes in the city of Pune, Maharashtra, India, killing 17 and injuring 60 more.
 * 2011 – For the first time in more than 100 years the Umatilla, an American Indian tribe, are able to hunt and harvest a bison just outside Yellowstone National Park, restoring a centuries-old tradition guaranteed by a treaty signed in 1855.
 * 2012 – The European Space Agency (ESA) conducted the first launch of the European Vega rocket from Europe's spaceport in Kourou, French Guiana.
 * 2017 – Kim Jong-nam, brother of North Korean dictator Kim Jong-un, is assassinated at Kuala Lumpur International Airport.
 * 2021 – Former U.S. President Donald Trump is acquitted in his second impeachment trial.
 * 2021  – A major winter storm causes blackouts and kills at least 82 people in Texas and northern Mexico.

Pre-1600

 * 1440 – Hartmann Schedel, German physician (d. 1514)
 * 1457 – Mary of Burgundy, Sovereign Duchess regnant of Burgundy, married to Maximilian I, Holy Roman Emperor (d. 1482)
 * 1469 – Elia Levita, Renaissance Hebrew grammarian (d. 1549)
 * 1480 – Girolamo Aleandro, Italian cardinal (d. 1542)
 * 1523 – Valentin Naboth, German astronomer and mathematician (d. 1593)
 * 1539 – Elisabeth of Hesse, Electress Palatine (d. 1582)
 * 1569 – Johann Reinhard I, Count of Hanau-Lichtenberg (d. 1625)
 * 1599 – Pope Alexander VII (d. 1667)

1601–1900

 * 1602 – William V, Landgrave of Hesse-Kassel (d. 1637)
 * 1672 – Étienne François Geoffroy, French physician and chemist (d. 1731)
 * 1683 – Giovanni Battista Piazzetta, Italian painter (d. 1754)
 * 1719 – George Brydges Rodney, 1st Baron Rodney, English admiral and politician (d. 1792)
 * 1721 – John Reid, Scottish general (d. 1807)
 * 1728 – John Hunter, Scottish surgeon and anatomist (d. 1793)
 * 1766 – Thomas Robert Malthus, English economist and scholar (d. 1834)
 * 1768 – Édouard Adolphe Casimir Joseph Mortier, French general and politician, 15th Prime Minister of France (d. 1835)
 * 1769 – Ivan Krylov, Russian author, poet, and playwright (d. 1844)
 * 1805 – Peter Gustav Lejeune Dirichlet, German mathematician and academic (d. 1859)
 * 1811 – François Achille Bazaine, French general (d. 1888)
 * 1815 – Rufus Wilmot Griswold, American anthologist, editor, poet and critic (d. 1857)
 * 1831 – John Aaron Rawlins, American general and politician, 29th United States Secretary of War (d. 1869)
 * 1834 – Heinrich Caro, Sephardic Jewish Polish-German chemist and academic (d. 1910)
 * 1835 – Mirza Ghulam Ahmad, Indian religious leader (d. 1908)
 * 1849 – Lord Randolph Churchill, English lawyer and politician, Chancellor of the Exchequer (d. 1895)
 * 1855 – Paul Deschanel, Belgian-French politician, 11th President of France (d. 1922)
 * 1863 – Hugo Becker, German cellist and composer (d. 1941)
 * 1867 – Harold Mahony, Scottish-Irish tennis player (d. 1905)
 * 1870 – Leopold Godowsky, Polish-American pianist and composer (d. 1938)
 * 1871 – Joseph Devlin, Northern Irish political leader (Nationalist Party (Northern Ireland)) (d. 1934)
 * 1873 – Feodor Chaliapin, Russian opera singer (d. 1938)
 * 1876 – Fritz Buelow, German-American baseball player and umpire (d. 1933)
 * 1879 – Sarojini Naidu, Indian poet and activist (d. 1949)
 * 1880 – Dimitrie Gusti, Romanian sociologist, ethnologist, historian, and philosopher (d. 1955)
 * 1881 – Eleanor Farjeon, English author, poet, and playwright (d. 1965)
 * 1883 – Hal Chase, American baseball player and manager (d. 1947)
 * 1883  – Yevgeny Vakhtangov, Russian-Armenian actor and director (d. 1922)
 * 1884 – Alfred Carlton Gilbert, American pole vaulter and businessman, founded the A. C. Gilbert Company (d. 1961)
 * 1885 – Bess Truman, 35th First Lady of the United States (d. 1982)
 * 1887 – Géza Csáth, Hungarian playwright and critic (d. 1919)
 * 1888 – Georgios Papandreou, Greek lawyer, economist, and politician, 162nd Prime Minister of Greece (d. 1968)
 * 1889 – Leontine Sagan, Austrian actress and director (d. 1974)
 * 1891 – Kate Roberts, Welsh author and activist (d. 1985)
 * 1891  – Grant Wood, American painter and academic (d. 1942)
 * 1892 – Robert H. Jackson, American lawyer, judge, and politician, 57th United States Attorney General (d. 1954)
 * 1898 – Hubert Ashton, English cricketer and politician (d. 1979)
 * 1899 – Rolf Stenersen, Norwegian businessman (d. 1978)
 * 1900 – Barbara von Annenkoff, Russian-born German film and stage actress (d. 1979)

1901–present

 * 1901 – Paul Lazarsfeld, Austrian-American sociologist and academic (d. 1976)
 * 1902 – Harold Lasswell, American political scientist and theorist (d. 1978)
 * 1903 – Georgy Beriev, Georgian-Russian engineer, founded the Beriev Design Bureau (d. 1979)
 * 1903  – Georges Simenon, Belgian-Swiss author (d. 1989)
 * 1906 – Agostinho da Silva, Portuguese philosopher and author (d. 1994)
 * 1907 – Katy de la Cruz, Filipino-American singer and actress (d. 2004)
 * 1910 – William Shockley, English-American physicist and academic, Nobel Prize laureate (d. 1989)
 * 1911 – Faiz Ahmad Faiz, Indian-Pakistani poet and journalist (d. 1984)
 * 1911  – Jean Muir, American actress and educator (d. 1996)
 * 1912 – Harald Riipalu, Russian-Estonian commander (d. 1961)
 * 1912  – Margaretta Scott, English actress (d. 2005)
 * 1913 – Khalid of Saudi Arabia (d. 1982)
 * 1915 – Lyle Bettger, American actor (d. 2003)
 * 1915  – Aung San, Burmese general and politician, 5th Premier of British Crown Colony of Burma (d. 1947)
 * 1916 – Dorothy Bliss, American invertebrate zoologist (d. 1987)
 * 1919 – Tennessee Ernie Ford, American singer and actor (d. 1991)
 * 1919  – Eddie Robinson, American football player and coach (d. 2007)
 * 1920 – Boudleaux Bryant, American songwriter (d. 1987)
 * 1920  – Eileen Farrell, American soprano and educator (d. 2002)
 * 1921 – Jeanne Demessieux, French pianist and composer (d. 1968)
 * 1921  – Aung Khin, Burmese painter (d. 1996)
 * 1922 – Francis Pym, Baron Pym, Welsh soldier and politician, Secretary of State for Foreign and Commonwealth Affairs (d. 2008)
 * 1922  – Gordon Tullock, American economist and academic (d. 2014)
 * 1923 – Michael Anthony Bilandic, American soldier, judge, and politician, 49th Mayor of Chicago (d. 2002)
 * 1923  – Chuck Yeager, American general and pilot; first test pilot to break the sound barrier (d. 2020)
 * 1924 – Jean-Jacques Servan-Schreiber, French journalist and politician (d. 2006)
 * 1926 – Fay Ajzenberg-Selove, American nuclear physicist (d. 2012)
 * 1928 – Gerald Regan, Canadian lawyer and politician, 19th Premier of Nova Scotia (d. 2019)
 * 1929 – Omar Torrijos, Panamanian commander and politician, Military Leader of Panama (d. 1981)
 * 1930 – Ernst Fuchs, Austrian painter, sculptor, and illustrator (d. 2015)
 * 1930  – Israel Kirzner, English-American economist, author, and academic
 * 1932 – Susan Oliver, American actress (d. 1990)
 * 1933 – Paul Biya, Cameroon politician, 2nd President of Cameroon
 * 1933  – Kim Novak, American actress
 * 1933  – Emanuel Ungaro, French fashion designer (d. 2019)
 * 1934 – George Segal, American actor (d. 2021)
 * 1937 – Ali El-Maak, Sudanese author and academic (d. 1992)
 * 1937  – Sigmund Jähn, German pilot and cosmonaut (d. 2019)
 * 1937  – Angelo Mosca, American-Canadian football player and wrestler (d. 2021)
 * 1938 – Oliver Reed, English actor (d. 1999)
 * 1940 – Bram Peper, Dutch sociologist and politician, Mayor of Rotterdam (d. 2022)
 * 1941 – Sigmar Polke, German painter and photographer (d. 2010)
 * 1941  – Bo Svenson, Swedish-American actor, director, and producer
 * 1942 – Carol Lynley, American model and actress (d. 2019)
 * 1942  – Peter Tork, American singer-songwriter, bass player, and actor (d. 2019)
 * 1942  – Donald E. Williams, American captain, pilot, and astronaut (d. 2016)
 * 1943 – Elaine Pagels, American theologian and academic
 * 1944 – Stockard Channing, American actress
 * 1944  – Jerry Springer, English-American television host, actor, and politician, 56th Mayor of Cincinnati (d. 2023)
 * 1945 – Marian Dawkins, English biologist and academic
 * 1945  – King Floyd, American singer-songwriter (d. 2006)
 * 1945  – Simon Schama, English historian and author
 * 1945  – William Sleator, American author and composer (d. 2011)
 * 1946 – Richard Blumenthal, American sergeant and politician, 23rd Attorney General of Connecticut
 * 1946  – Janet Finch, English sociologist and academic
 * 1946  – Colin Matthews, English composer and educator
 * 1947 – Stephen Hadley, American soldier and diplomat, 21st United States National Security Advisor
 * 1947  – Mike Krzyzewski, American basketball player and coach
 * 1947  – Bogdan Tanjević, Montenegrin-Bosnian basketball coach
 * 1947  – Kevin Bloody Wilson, Australian comedian, singer-songwriter, and guitarist
 * 1949 – Peter Kern, Austrian actor, director, producer, and screenwriter (d. 2015)
 * 1950 – Vera Baird, English lawyer and politician
 * 1950  – Peter Gabriel, English singer-songwriter and musician
 * 1951 – David Naughton, American actor and singer
 * 1952 – Ed Gagliardi, American musician (d. 2014)
 * 1953 – Akio Sato, Japanese wrestler and manager
 * 1954 – Donnie Moore, American baseball player (d. 1989)
 * 1955 – Joe Birkett, American lawyer, judge, and politician
 * 1956 – Peter Hook, English singer, songwriter, musician, and record producer
 * 1957 – Denise Austin, American fitness trainer and author
 * 1958 – Pernilla August, Swedish actress
 * 1958  – Øivind Elgenes, Norwegian singer, guitarist, and composer
 * 1958  – Marc Emery, Canadian publisher and activist
 * 1958  – Jean-François Lisée, Canadian journalist and politician
 * 1958  – Derek Riggs, English painter and illustrator
 * 1959 – Gaston Gingras, Canadian ice hockey player
 * 1960 – Pierluigi Collina, Italian footballer and referee
 * 1960  – John Healey, English journalist and politician
 * 1960  – Gary Patterson, American football player and coach
 * 1960  – Matt Salinger, American actor
 * 1960  – Artur Yusupov, Russian-German chess player and author
 * 1961 – Marc Crawford, Canadian ice hockey player and coach
 * 1961  – cEvin Key, Canadian singer-songwriter, drummer, keyboard player, and producer
 * 1961  – Henry Rollins, American singer-songwriter, producer, and actor
 * 1962 – Aníbal Acevedo Vilá, Puerto Rican lawyer and politician
 * 1962  – Baby Doll, American wrestler and manager
 * 1962  – Michele Greene, American actress
 * 1964 – Stephen Bowen, American engineer, captain, and astronaut
 * 1964  – Ylva Johansson, Swedish educator and politician, Swedish Minister of Employment
 * 1965 – Peter O'Neill, Papua New Guinean accountant and politician, 7th Prime Minister of Papua New Guinea
 * 1966 – Neal McDonough, American actor and producer
 * 1966  – Jeff Waters, Canadian guitarist, songwriter, and producer
 * 1966  – Freedom Williams, American rapper and singer
 * 1967 – Stanimir Stoilov, Bulgarian footballer and coach
 * 1968 – Kelly Hu, American actress
 * 1969 – Joyce DiDonato, American soprano and actress
 * 1969  – Bryan Thomas Schmidt, American science fiction author and editor
 * 1970 – Elmer Bennett, American basketball player
 * 1970  – Karoline Krüger, Norwegian singer-songwriter and pianist
 * 1971 – Sonia Evans, English singer-songwriter
 * 1971  – Mats Sundin, Swedish ice hockey player
 * 1971  – Todd Williams, American baseball player
 * 1972 – Virgilijus Alekna, Lithuanian discus thrower
 * 1972  – Juha Ylönen, Finnish ice hockey player
 * 1974 – Fonzworth Bentley, American rapper and actor
 * 1974  – Robbie Williams, English singer-songwriter
 * 1975 – Ben Collins, English race car driver
 * 1975  – Katie Hopkins, English media personality and columnist
 * 1976 – Jörg Bergmeister, German race car driver
 * 1976  – Feist, Canadian singer-songwriter and musician
 * 1977 – Randy Moss, American football player and coach
 * 1978 – Niklas Bäckström, Finnish ice hockey player
 * 1978  – Philippe Jaroussky, French singer
 * 1978  – Cory Murphy, Canadian ice hockey player and coach
 * 1979 – Anders Behring Breivik, Norwegian murderer
 * 1979  – Rafael Márquez, Mexican footballer
 * 1979  – Rachel Reeves, English economist and politician, Shadow Chancellor of the Exchequer
 * 1979  – Mena Suvari, American actress and fashion designer
 * 1980 – Carlos Cotto, Puerto Rican-American wrestler and boxer
 * 1981 – Luisão, Brazilian footballer
 * 1981  – Luke Ridnour, American basketball player
 * 1982 – Even Helte Hermansen, Norwegian guitarist and composer
 * 1982  – Michael Turner, American football player
 * 1983 – Mike Nickeas, Canadian baseball player
 * 1983  – Anna Watkins, English rower
 * 1984 – Hinkelien Schreuder, Dutch swimmer
 * 1985 – Somdev Devvarman, Indian tennis player
 * 1985  – J. R. Giddens, American basketball player
 * 1985  – Kwak Ji-min, South Korean actress
 * 1985  – Al Montoya, American ice hockey player
 * 1986 – Luke Moore, English footballer
 * 1986  – Aqib Talib, American football player
 * 1987 – Eljero Elia, Dutch footballer
 * 1988 – Ryan Goins, American baseball player
 * 1988  – Dave Rudden, Irish author
 * 1989 – Rodrigo Possebon, Brazilian footballer
 * 1990 – Nathan Eovaldi, American baseball player
 * 1990  – Mamadou Sakho, French footballer
 * 1991 – Eliaquim Mangala, French footballer
 * 1991  – Vianney, French singer
 * 1991  – Luke Voit, American baseball player
 * 1992 – Keith Appling, American basketball player
 * 1994 – Memphis Depay, Dutch footballer
 * 1995 – Kendall Fuller, American football player
 * 1995  – Georges-Kévin Nkoudou, French-Cameroonian footballer
 * 2000 – Vitinha, Portuguese footballer
 * 2001 – Kaapo Kakko, Finnish ice hockey player
 * 2002 – Jaden Ivey, American basketball player
 * 2002  – Sophia Lillis, American actress

Pre-1600

 * 106 – Emperor He of Han (Han Hedi) of the Chinese Eastern Han Dynasty (b. AD 79)
 * 721 – Chilperic II, Frankish king (b. 672)
 * 858 – Kenneth MacAlpin, Scottish king (probable; b. 810)
 * 921 – Vratislaus I, duke of Bohemia
 * 936 – Xiao Wen, empress of the Liao Dynasty
 * 942 – Muhammad ibn Ra'iq, Abbasid emir and regent
 * 988 – Adalbert Atto, Lombard nobleman
 * 1021 – Al-Hakim bi-Amr Allah, Fatimid caliph (b. 985)
 * 1130 – Honorius II, pope of the Catholic Church (b. 1060)
 * 1141 – Béla II, king of Hungary and Croatia (b. 1110)
 * 1199 – Stefan Nemanja, Serbian grand prince (b. 1113)
 * 1219 – Minamoto no Sanetomo, Japanese shōgun (b. 1192)
 * 1332 – Andronikos II Palaiologos, Byzantine emperor (b. 1259)
 * 1351 – Kō no Morofuyu, Japanese general
 * 1539 – Isabella d'Este, Italian noblewoman (b. 1474)
 * 1542 – Catherine Howard, English wife of Henry VIII of England (executed; b. 1521)
 * 1571 – Benvenuto Cellini, Italian painter and sculptor (b. 1500)
 * 1585 – Alfonso Salmeron, Spanish priest and scholar (b. 1515)

1601–1900

 * 1602 – Alexander Nowell, English clergyman and theologian (b. 1507)
 * 1660 – Charles X Gustav, king of Sweden (b. 1622)
 * 1662 – Elizabeth Stuart, queen of Bohemia (b. 1596)
 * 1693 – Johann Caspar Kerll, German organist and composer (b. 1627)
 * 1727 – William Wotton, English linguist and scholar (b. 1666)
 * 1728 – Cotton Mather, American minister and author (b. 1663)
 * 1732 – Charles-René d'Hozier, French historian and author (b. 1640)
 * 1741 – Johann Joseph Fux, Austrian composer and theorist (b. 1660)
 * 1787 – Roger Joseph Boscovich, Croatian physicist, astronomer, mathematician, and philosopher (b. 1711)
 * 1787  – Charles Gravier, comte de Vergennes, French lawyer and politician, Foreign Minister of France (b. 1717)
 * 1795 – George (Konissky), Orthodox archbishop, preacher, philosopher and theologian (b. 1717)
 * 1813 – Samuel Ashe, American lawyer and politician, 9th Governor of North Carolina (b. 1725)
 * 1818 – George Rogers Clark, American general (b. 1752)
 * 1826 – Peter Ludwig von der Pahlen, Russian general and politician, Governor-General of Baltic provinces (b. 1745)
 * 1831 – Edward Berry, English admiral (b. 1768)
 * 1837 – Mariano José de Larra, Spanish journalist and author (b. 1809)
 * 1845 – Henrik Steffens, Norwegian-German philosopher and poet (b. 1773)
 * 1859 – Eliza Acton, English food writer and poet (b. 1799)
 * 1877 – Costache Caragiale, Romanian actor and manager (b. 1815)
 * 1883 – Richard Wagner, German composer (b. 1813)
 * 1888 – Jean-Baptiste Lamy, French-American archbishop (b. 1814)
 * 1892 – Provo Wallis, Canadian-English admiral (b. 1791)
 * 1893 – Ignacio Manuel Altamirano, Mexican intellectual and journalist (b. 1834)

1901–present

 * 1905 – Konstantin Savitsky, Russian painter (b. 1844)
 * 1906 – Albert Gottschalk, Danish painter (b. 1866)
 * 1934 – József Pusztai, Slovene-Hungarian poet and journalist (b. 1864)
 * 1942 – Otakar Batlička, Czech journalist (b. 1895)
 * 1942  – Epitácio Pessoa, Brazilian lawyer, judge, and politician, 11th President of Brazil (b. 1865)
 * 1950 – Rafael Sabatini, Italian-English novelist and short story writer (b. 1875)
 * 1951 – Lloyd C. Douglas, American minister and author (b. 1877)
 * 1952 – Josephine Tey, Scottish author and playwright (b. 1896)
 * 1954 – Agnes Macphail, Canadian educator and politician (b. 1890)
 * 1956 – Jan Łukasiewicz, Polish mathematician and philosopher (b. 1878)
 * 1958 – Christabel Pankhurst, English activist, co-founded the Women's Social and Political Union (b. 1880)
 * 1958  – Georges Rouault, French painter and illustrator (b. 1871)
 * 1964 – Paulino Alcántara, Filipino-Spanish footballer and manager (b. 1896)
 * 1964  – Werner Heyde, German psychiatrist and academic (b. 1902)
 * 1967 – Yoshisuke Aikawa, entrepreneur, businessman, and politician, founded Nissan Motor Company (b. 1880)
 * 1967  – Abelardo L. Rodríguez, substitute president of Mexico (1932-1934) (b. 1889)
 * 1968 – Mae Marsh, American actress (b. 1895)
 * 1968  – Portia White, Canadian opera singer (b. 1911)
 * 1973 – Marinus Jan Granpré Molière, Dutch architect and educator (b. 1883)
 * 1975 – André Beaufre, French general (b. 1902)
 * 1976 – Murtala Mohammed, Nigerian general and politician, 4th President of Nigeria (b. 1938)
 * 1976  – Lily Pons, French-American soprano and actress (b. 1904)
 * 1980 – David Janssen, American actor (b. 1931)
 * 1984 – Cheong Eak Chong, Singaporean entrepreneur (b. 1888)
 * 1986 – Yuri Ivask, Russian-American poet and critic (b. 1907)
 * 1989 – Wayne Hays, American lieutenant and politician (b. 1911)
 * 1991 – Arno Breker, German sculptor and illustrator (b. 1900)
 * 1992 – Nikolay Bogolyubov, Ukrainian-Russian mathematician and physicist (b. 1909)
 * 1996 – Martin Balsam, American actor (b. 1919)
 * 1997 – Robert Klark Graham, American eugenicist and businessman (b. 1906)
 * 1997  – Mark Krasnosel'skii, Russian-Ukrainian mathematician and academic (b. 1920)
 * 2000 – Anders Aalborg, Canadian educator and politician (b. 1914)
 * 2000  – James Cooke Brown, American sociologist and author (b. 1921)
 * 2000  – John Leake, English soldier (b. 1949)
 * 2002 – Waylon Jennings, American singer-songwriter and guitarist (b. 1937)
 * 2003 – Kid Gavilán, Cuban-American boxer (b. 1926)
 * 2003  – Walt Whitman Rostow, American economist; 7th United States National Security Advisor (b. 1916)
 * 2004 – François Tavenas, Canadian engineer and academic (b. 1942)
 * 2004  – Zelimkhan Yandarbiyev, Chechen politician, 2nd President of the Chechen Republic of Ichkeria (b. 1952)
 * 2005 – Nelson Briles, American baseball player and sportscaster (b. 1943)
 * 2005  – Lúcia Santos, Portuguese nun (b. 1907)
 * 2006 – P. F. Strawson, English philosopher and author (b. 1919)
 * 2007 – Elizabeth Jolley, English-Australian author and academic (b. 1923)
 * 2007  – Charlie Norwood, American captain and politician (b. 1941)
 * 2007  – Richard Gordon Wakeford, English air marshal (b. 1922)
 * 2008 – Kon Ichikawa, Japanese director, producer, and screenwriter (b. 1915)
 * 2009 – Edward Upward, English author and educator (b. 1903)
 * 2010 – Lucille Clifton, American poet and academic (b. 1936)
 * 2010  – Dale Hawkins, American singer-songwriter and guitarist (b. 1936)
 * 2012 – Russell Arms, American actor and singer (b. 1920)
 * 2012  – Louise Cochrane, American-English screenwriter and producer (b. 1918)
 * 2012  – Daniel C. Gerould, American playwright and academic (b. 1928)
 * 2013 – Gerry Day, American journalist and screenwriter (b. 1922)
 * 2013  – Miles J. Jones, American pathologist and physician (b. 1952)
 * 2013  – Pieter Kooijmans, Dutch judge and politician, Minister of Foreign Affairs for The Netherlands (b. 1933)
 * 2013  – Andrée Malebranche, Haitian artist (b. 1916)
 * 2013  – Yuko Tojo, Japanese activist and politician (b. 1939)
 * 2014 – Balu Mahendra, Sri Lankan-Indian director, cinematographer, and screenwriter (b. 1939)
 * 2014  – Richard Møller Nielsen, Danish footballer and manager (b. 1937)
 * 2014  – Ralph Waite, American actor and activist (b. 1928)
 * 2015 – Faith Bandler, Australian activist and author (b. 1918)
 * 2015  – Stan Chambers, American journalist and actor (b. 1923)
 * 2016 – O. N. V. Kurup, Indian poet and academic (b. 1931)
 * 2016  – Antonin Scalia, American lawyer and judge, Associate Justice of the Supreme Court of the United States (b. 1936)
 * 2017 – Ricardo Arias Calderón, Panamanian politician (b. 1933)
 * 2017  – Aileen Hernandez, American union organizer and activist (b. 1926)
 * 2017  – Seijun Suzuki, Japanese filmmaker (b. 1923)
 * 2017  – Kim Jong-nam, North Korean politician (b. 1971)
 * 2017  – E-Dubble, American rapper (b. 1982)
 * 2018 – Henrik, Prince Consort of Denmark, French-born Danish royal (b. 1934)
 * 2019 – Callistus Ndlovu, Zimbabwean academic and politician (b. 1936)
 * 2021 – Kadir Topbaş, Turkish politician (b. 1945)

Holidays and observances

 * Children's Day (Myanmar)
 * Christian feast day:
 * Absalom Jones (Episcopal Church (USA))
 * Beatrice of Ornacieux
 * Castor of Karden
 * Catherine of Ricci
 * Dyfnog
 * Ermenilda of Ely
 * Fulcran
 * Jordan of Saxony
 * Polyeuctus (Roman Catholic Church)
 * February 13 (Eastern Orthodox liturgics)
 * World Radio Day

title: February 13


text:
Talk:Fundamental interaction

relative strength and range
The strong interaction has a relative strength of 1038, but a small range of 10-15 m. If the relative strength of 1038 had infinite range, the strong interaction would pull everything in and make 1 giant atomic nucleus.

But, what if the small range of 10-15 m had infinite relative strength? 84.154.74.115 (talk) 20:44, 20 April 2023 (UTC)

Jugs of water example
So, not a physicist here, Is there any reference for this jugs of water example? Also how is this even possible there is probably no way to make two jugs of water with the exact amount of water and the example claims that the electromagnetic forces between the two just cancel each other out. I'm not sure if the example is just not correct or what is going for anyone who knows more physics that I do please take a look and maybe replace with a believable example. — Preceding unsigned comment added by Goodmanguy (talk • contribs) 05:21, 9 December 2023 (UTC)

 * Feel free to delete that paragraph: as you say it is unreferenced which means any editor can challenge it.
 * All the example is saying is that neutral atoms have no force between them despite the strong electromagnetic force between the electrons themselves and the protons themselves: each atom has an matching number of each charge. Multiply by the number of atoms in two jugs of water, same result: zero times any number still zero. The jugs part just makes the electron-electron repulsion large.
 * (There is a small London force between neutral atoms). Johnjbarton (talk) 16:26, 9 December 2023 (UTC)

heat death
my guess for 3 Ways to Destroy the Universe

Over large (astronomical) distances, gravity tends to be the dominant force, and is responsible for holding together the large scale structures in the universe, such as planets, stars, and galaxies. However, gravity is the weakest of the four interactions. With these 2 properties of gravity, I guess, the death of the universe will be the heat death. 94.31.85.138 (talk) 16:08, 9 December 2023 (UTC)

Gravity is not a force
Just a simple Google search produces results here; Dr Sabine Hossenfelder recently put out an explainer on YouTube.

This is a complex topic and referring to a YouTube video feels "un-academic," however myriad Google and Google Scholar results lead you to numerous explainers on why it is not a force. Thus, it is disingenuous to keep this article as it currently stands. My ability to convey this complex topic is limited, so I defer to more physics-minded users to rewrite the page, but my point stands: Gravity is not a force. 38.74.24.37 (talk) 05:55, 29 December 2023 (UTC)

 * As a physics-minded user, I will point out that the name of this page is "Fundamental interaction", a term used in part to avoid such pointless discussions. It makes no sense to say "gravity is not a force" unless you define "force", which as it turns out is quite a complex and ambiguous business. F=mgh is one definition of force and the "g" there clearly makes gravity a force. This particular definition is wonderfully useful and I recommend it to you as a topic of study: Newton's law of universal gravitation. Hossenfelder's video is also not incorrect, but rather a sensationalization of the topic, choosing to emphasize the general relativity field-theory description of gravitation. Also a wonderful topic to learn more about. In that theory gravity appears as a consequence of 4D geometry, but you might think about what definition of "force" means in that geometry.
 * I don't consider this youtube video to be a valuable reference on the article topic. Nevertheless I suppose this article could call out the issue of force more directly. Johnjbarton (talk) 17:48, 29 December 2023 (UTC)
 * As I said, using YouTube is not an academic source. However, when academics (such as Dr Sabine Hossenfelder) post highlights of their and others' research, it becomes a valuable tool to jump off into deeper research, thus me saying that Google and, especially, Google Scholar maintain rigorous research and writing on this very topic.
 * This said, thank you for the informed response! My background is linguistics rather than physics, so I appreciate the input and would urge more clarity in the live article. 38.74.24.37 (talk) 15:58, 30 December 2023 (UTC)
 * (Edit: YouTube can be an academic source as video essays with rigorously sourced materials and well-written research are valuable, I would like to note) 38.74.24.37 (talk) 16:00, 30 December 2023 (UTC)
 * I'm not against youtube as a source, esp. for External links. In general however youtube is not considered a reliable source because the videos are not peer-reviewed nor typically cited by other authors. In this particular case, my reasons against the video is that it is off-topic: it is about general relativity and does not speak about gravity as one of the fundamental interactions in physics.
 * BTW a sizable fraction of the physics community believes that a theory of quantum gravity will eventually show that the underlying mechanism of gravity is the same as the other three interactions. Such a theory would not alter the curvature of spacetime aspect of gravity just give a different way to calculate it. Johnjbarton (talk) 16:28, 30 December 2023 (UTC)

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

title: Talk:Fundamental interaction


text:
Political freedom

Political freedom (also known as political autonomy or political agency) is a central concept in history and political thought and one of the most important features of democratic societies. Political freedom has been described as freedom from oppression or coercion, the absence of disabling conditions for an individual and the fulfillment of enabling conditions, or the absence of life conditions of compulsion, e.g. economic compulsion, in a society. Although political freedom is often interpreted negatively as the freedom from unreasonable external constraints on action, it can also refer to the positive exercise of rights, capacities and possibilities for action and the exercise of social or group rights. The concept can also include freedom from internal constraints on political action or speech (e.g. social conformity, consistency, or inauthentic behaviour). The concept of political freedom is closely connected with the concepts of civil liberties and human rights, which in democratic societies are usually afforded legal protection from the state.

Views
Various groups along the political spectrum hold different views about what they believe constitutes political freedom.

Left-wing political philosophy generally couples the notion of freedom with that of positive liberty or the enabling of a group or individual to determine their own life or realize their own potential. In this sense, freedom may include freedom from poverty, starvation, treatable disease, and oppression as well as freedom from force and coercion, from whomever they may issue.

According to neoliberal philosopher and economist Friedrich Hayek, the "socialist argument" defined "individual liberty" as " 'freedom from' obstacles". He argued that this definition only "confused" and obscured the aim of "securing individual freedom", because it permitted a possible "identification of freedom with power." The subsequent "collective power over circumstances" misappropriated "the physical 'ability to do what I want', the power to satisfy our wishes, or the extent of the choice of alternatives open to us." Hayek maintained that once any possible "identification of freedom with power is admitted," a "totalitarian state" coalesced where "liberty has been suppressed in the name of liberty."

Social anarchists see negative and positive liberty as complementary concepts of freedom. Such a view of rights may require utilitarian trade-offs, such as sacrificing the right to the product of one's labor or freedom of association for less racial discrimination or more subsidies for housing. Social anarchists describe the negative liberty-centric view endorsed by capitalism as "selfish freedom".

Anarcho-capitalists see negative rights as a consistent system. Ayn Rand described it as "a moral principle defining and sanctioning a man's freedom of action in a social context". To such libertarians, positive liberty is contradictory since so-called rights must be traded off against each other, debasing legitimate rights which by definition trump other moral considerations. Any alleged right which calls for an end result (e.g. housing, education, medical services and so on) produced by people is in effect a purported right to enslave others.

Political philosopher Alasdair MacIntyre theorized freedom in terms of our social interdependence with other people.

Economist Milton Friedman argues in his book Capitalism and Freedom that there are two types of freedom, namely political freedom and economic freedom, and that without economic freedom there cannot be political freedom.

A study on 123 nations shows that the Hayek–Friedman hypothesis of the necessity of economic freedom for political freedom holds well. Only in few cases, there was relatively high political freedom without high economic freedom, and even those cases diminished over time. A later study found just one clear counter-example, Belarus after 1991, and its freedom was lost over time.

In his article "Why the Market Subverts Democracy", Robin Hahnel takes issue with Friedman's concept of economic freedom, asserting that there will be infringements on the freedom of others whenever anyone exercises their own economic freedom. He argues that such infringements produce conflicts that are resolved through property rights systems, and therefore it is essential to decide what is a better or a worse property rights system, yet Friedman simply takes for granted the existing property rights and does not question them.

Political philosopher Nikolas Kompridis posits that the pursuit of freedom in the modern era can be broadly divided into two motivating ideals, namely freedom as autonomy or independence and freedom as the ability to cooperatively initiate a new beginning.

Political freedom has also been theorized in its opposition to and a condition of power relations, or the power of action upon actions, by Michel Foucault. It has also been closely identified with certain kinds of artistic and cultural practice by Cornelius Castoriadis, Antonio Gramsci, Herbert Marcuse, Jacques Rancière and Theodor Adorno.

Environmentalists often argue that political freedoms should include some constraint on use of ecosystems. They maintain there is no such thing, for instance, as freedom to pollute or freedom to deforest given that such activities create negative externalities, which violates other groups' liberty to not be exposed to pollution. The popularity of SUVs, golf and urban sprawl has been used as evidence that some ideas of freedom and ecological conservation can clash. This leads at times to serious confrontations and clashes of values reflected in advertising campaigns, e.g. that of PETA regarding fur.

John Dalberg-Acton stated: "The most certain test by which we judge whether a country is really free is the amount of security enjoyed by minorities."

Gerald C. MacCallum Jr. spoke of a compromise between positive and negative freedoms, saying that an agent must have full autonomy over themselves. In this view, freedom is a triadic relationship because it is about three things, namely the agent, the constraints they need to be free from and the goal they are aspiring to.

History
Hannah Arendt traces the conceptual origins of freedom to ancient Greek politics. According to her study, the concept of freedom was historically inseparable from political action. Politics could only be practiced by those who had freed themselves from the necessities of life so that they could participate in the realm of political affairs. According to Arendt, the concept of freedom became associated with the Christian notion of freedom of the will, or inner freedom, around the 5th century CE and since then freedom as a form of political action has been neglected even though, as she says, freedom is "the raison d'être of politics".

Arendt says that political freedom is historically opposed to sovereignty or will-power since in ancient Greece and Rome the concept of freedom was inseparable from performance and did not arise as a conflict between the will and the self. Similarly, the idea of freedom as freedom from politics is a notion that developed in modern times. This is opposed to the idea of freedom as the capacity to "begin anew", which Arendt sees as a corollary to the innate human condition of natality, or our nature as "new beginnings and hence beginners".

In Arendt's view, political action is an interruption of automatic process, either natural or historical. The freedom to begin anew is thus an extension of "the freedom to call something into being which did not exist before, which was not given, not even as an object of cognition or imagination, and which therefore, strictly speaking, could not be known".

title: Political freedom


text:
Foobar

The terms foobar, foo, bar, baz, qux, quux, and others are used as metasyntactic variables and placeholder names in computer programming or computer-related documentation. They have been used to name entities such as variables, functions, and commands whose exact identity is unimportant and serve only to demonstrate a concept. The style guide for Google developer documentation recommends against using them as example project names because they are unclear and can cause confusion.

History and etymology
It is possible that foobar is a playful allusion to the World War II-era military slang FUBAR (Fucked Up Beyond All Recognition).

Note however that the word foopaw was a Nantucket dialect word for a bungled job, apparently derived from the French faux pas, quoted from William H. Macy's '"There She Blows - The Log of the Arethusa" (1877).

According to an Internet Engineering Task Force RFC, the word FOO originated as a nonsense word with its earliest documented use in the 1930s comic Smokey Stover by Bill Holman. Holman states that he used the word due to having seen it on the bottom of a jade Chinese figurine in San Francisco Chinatown, purportedly signifying "good luck". If true, this is presumably related to the Chinese word fu ("福", sometimes transliterated foo, as in foo dog), which can mean happiness or blessing.

The first known use of the terms in print in a programming context appears in a 1965 edition of MIT's Tech Engineering News. The use of foo in a programming context is generally credited to the Tech Model Railroad Club (TMRC) of MIT from c. 1960. In the complex model system, there were scram switches located at numerous places around the room that could be thrown if something undesirable was about to occur, such as a train moving at full power towards an obstruction. Another feature of the system was a digital clock on the dispatch board. When someone hit a scram switch, the clock stopped and the display was replaced with the word "FOO"; at TMRC the scram switches are, therefore, called "Foo switches". Because of this, an entry in the 1959 Dictionary of the TMRC Language went something like this: "FOO: The first syllable of the misquoted sacred chant phrase 'foo mane padme hum.' Our first obligation is to keep the foo counters turning." One book describing the MIT train room describes two buttons by the door labeled "foo" and "bar". These were general-purpose buttons and were often repurposed for whatever fun idea the MIT hackers had at the time, hence the adoption of foo and bar as general-purpose variable names. An entry in the Abridged Dictionary of the TMRC Language states:

"Multiflush: stop-all-trains-button. Next best thing to the red door button. Also called FOO. Displays 'FOO' on the clock when used."

Foobar was used as a variable name in the Fortran code of Colossal Cave Adventure (1977 Crowther and Woods version). The variable FOOBAR was used to contain the player's progress in saying the magic phrase "Fee Fie Foe Foo", a phrase from an historical quatrain in the classic English fairy tale Jack and the Beanstalk. Intel also used the term foo in their programming documentation in 1978.

Examples in culture

 * Foo Camp is an annual hacker convention.
 * BarCamp, an international network of user-generated conferences
 * During the United States v. Microsoft Corp. trial, evidence was presented that Microsoft had tried to use the Web Services Interoperability organization (WS-I) as a means to stifle competition, including e-mails in which top executives including Bill Gates and Steve Ballmer referred to the WS-I using the codename "foo".
 * foobar2000 is an audio player.

title: Foobar


text:
Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, or topology) and the linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous or unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

The usage of the word functional as a noun goes back to the calculus of variations, implying a function whose argument is a function. The term was first used in Hadamard's 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by the Italian mathematician and physicist Vito Volterra. The theory of nonlinear functionals was continued by students of Hadamard, in particular Fréchet and Lévy. Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach.

In modern introductory texts on functional analysis, the subject is seen as the study of vector spaces endowed with a topology, in particular infinite-dimensional spaces. In contrast, linear algebra deals mostly with finite-dimensional spaces, and does not use topology. An important part of functional analysis is the extension of the theories of measure, integration, and probability to infinite dimensional spaces, also known as infinite dimensional analysis.

Normed vector spaces
The basic and historically first class of spaces studied in functional analysis are complete normed vector spaces over the real or complex numbers. Such spaces are called Banach spaces. An important example is a Hilbert space, where the norm arises from an inner product. These spaces are of fundamental importance in many areas, including the mathematical formulation of quantum mechanics, machine learning, partial differential equations, and Fourier analysis.

More generally, functional analysis includes the study of Fréchet spaces and other topological vector spaces not endowed with a norm.

An important object of study in functional analysis are the continuous linear operators defined on Banach and Hilbert spaces. These lead naturally to the definition of C*-algebras and other operator algebras.

Hilbert spaces
Hilbert spaces can be completely classified: there is a unique Hilbert space up to isomorphism for every cardinality of the orthonormal basis. Finite-dimensional Hilbert spaces are fully understood in linear algebra, and infinite-dimensional separable Hilbert spaces are isomorphic to $\ell^{\,2}(\aleph_0)\,$. Separability being important for applications, functional analysis of Hilbert spaces consequently mostly deals with this space. One of the open problems in functional analysis is to prove that every bounded linear operator on a Hilbert space has a proper invariant subspace. Many special cases of this invariant subspace problem have already been proven.

Banach spaces
General Banach spaces are more complicated than Hilbert spaces, and cannot be classified in such a simple manner as those. In particular, many Banach spaces lack a notion analogous to an orthonormal basis.

Examples of Banach spaces are $L^p$-spaces for any real number $p\geq1$. Given also a measure $$\mu$$ on set $X$, then $L^p(X)$, sometimes also denoted $$L^p(X,\mu)$$ or $L^p(\mu)$, has as its vectors equivalence classes $$[\,f\,]$$ of measurable functions whose absolute value's $$p$$-th power has finite integral; that is, functions $$f$$ for which one has $$\int_{X}\left|f(x)\right|^p\,d\mu(x) < \infty.$$

If $$\mu$$ is the counting measure, then the integral may be replaced by a sum. That is, we require $$\sum_{x\in X}\left|f(x)\right|^p < \infty .$$

Then it is not necessary to deal with equivalence classes, and the space is denoted $\ell^p(X)$, written more simply $$\ell^p$$ in the case when $$X$$ is the set of non-negative integers.

In Banach spaces, a large part of the study involves the dual space: the space of all continuous linear maps from the space into its underlying field, so-called functionals. A Banach space can be canonically identified with a subspace of its bidual, which is the dual of its dual space. The corresponding map is an isometry but in general not onto. A general Banach space and its bidual need not even be isometrically isomorphic in any way, contrary to the finite-dimensional situation. This is explained in the dual space article.

Also, the notion of derivative can be extended to arbitrary functions between Banach spaces. See, for instance, the Fréchet derivative article.

Major and foundational results
There are four major theorems which are sometimes called the four pillars of functional analysis:

 * the Hahn–Banach theorem
 * the open mapping theorem
 * the closed graph theorem
 * the uniform boundedness principle, also known as the Banach–Steinhaus theorem.

Important results of functional analysis include:

Uniform boundedness principle
The uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm.

The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus but it was also proven independently by Hans Hahn.

$$

Spectral theorem
There are many theorems known as the spectral theorem, but one in particular has many applications in functional analysis.

$$

This is the beginning of the vast research area of functional analysis called operator theory; see also the spectral measure.

There is also an analogous spectral theorem for bounded normal operators on Hilbert spaces. The only difference in the conclusion is that now $$f$$ may be complex-valued.

Hahn–Banach theorem
The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting".

$$

Open mapping theorem
The open mapping theorem, also known as the Banach–Schauder theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map. More precisely, $$

The proof uses the Baire category theorem, and completeness of both $$X$$ and $$Y$$ is essential to the theorem. The statement of the theorem is no longer true if either space is just assumed to be a normed space, but is true if $$X$$ and $$Y$$ are taken to be Fréchet spaces.

Closed graph theorem
The closed graph theorem states the following: If $$X$$ is a topological space and $$Y$$ is a compact Hausdorff space, then the graph of a linear map $$T$$ from $$X$$ to $$Y$$ is closed if and only if $$T$$ is continuous.

Foundations of mathematics considerations
Most spaces considered in functional analysis have infinite dimension. To show the existence of a vector space basis for such spaces may require Zorn's lemma. However, a somewhat different concept, the Schauder basis, is usually more relevant in functional analysis. Many theorems require the Hahn–Banach theorem, usually proved using the axiom of choice, although the strictly weaker Boolean prime ideal theorem suffices. The Baire category theorem, needed to prove many important theorems, also requires a form of axiom of choice.

Points of view
Functional analysis includes the following tendencies:
 * Abstract analysis. An approach to analysis based on topological groups, topological rings, and topological vector spaces.
 * Geometry of Banach spaces contains many topics. One is combinatorial approach connected with Jean Bourgain; another is a characterization of Banach spaces in which various forms of the law of large numbers hold.
 * Noncommutative geometry. Developed by Alain Connes, partly building on earlier notions, such as George Mackey's approach to ergodic theory.
 * Connection with quantum mechanics. Either narrowly defined as in mathematical physics, or broadly interpreted by, for example, Israel Gelfand, to include most types of representation theory.

title: Functional analysis