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Brian Jacques

James Brian Jacques (, as in "Jakes"; 15 June 1939 – 5 February 2011) was an English author known for his Redwall series of novels and Castaways of the Flying Dutchman series. He also completed two collections of short stories entitled The Ribbajack & Other Curious Yarns and Seven Strange and Ghostly Tales.

Early life
James Brian Jacques was born in Liverpool on 15 June 1939. Jacques' parents were James Alfred Jacques, a truck driver and Ellen Ryan, both born in Liverpool. His father's family were from Lancashire, his mother's family all had Irish roots. Jacques' maternal grandfather, Matthew Ryan, was from Wexford, Ireland. Jacques was the middle child: he had an older brother, Tony, and a younger brother, James.

Jacques grew up in Kirkdale near to the Liverpool Docks. He was known by his middle name, Brian, because his father and younger brother were also named James. His father loved literature and read his boys adventure stories by Daniel Defoe, Sir Thomas Mallory, Sir Arthur Conan Doyle, Robert Louis Stevenson, and Edgar Rice Burroughs, but also The Wind in the Willows with its cast of animals. Jacques showed early writing talent.

At age ten, assigned to write an animal story, he wrote about a bird that cleaned a crocodile's teeth. His teacher could not believe that a ten-year-old wrote it, and caned him for refusing to admit copying the story. He attended St John's Roman Catholic school in Kirkdale, where his favourite teacher was Austin Thomas, a former Second World War army captain. Thomas had a major impact on Jacques: "I was fourteen at the time when Mr. Thomas introduced the class to poetry and Greek literature. It was because of him, I saved seven shillings and sixpence to buy The Iliad and The Odyssey at this dusty used book shop."

Career
Jacques left school at age fifteen, as was usual at the time, and set out to find adventure as a merchant sailor. When he returned to Liverpool, he began a varied career, spending time as a railway fireman, longshoreman, long-distance truck driver, bus driver, postmaster, and a stand-up entertainer. However, he often visited the local public library to continue his love of reading, and continued to develop his writing abilities. He published a succession of humorous poems and short stories through the 1970s, and in 1981 won a long term Residency at the Everyman Theatre, Liverpool, where his plays Brown Bitter, Wet Nellies and Scouse were performed.

In the 1980s, Jacques worked as a milkman, on a round which included the Royal Wavertree School for the Blind. He got to know the children there, and volunteered to read to them. However, he became dissatisfied with the state of children's literature, with too much adolescent angst, and began to write stories for them. So that the visually impaired children would be able to picture the scenes he was writing for them, he developed a highly descriptive style, emphasizing sound, smell, taste, gravity, balance, temperature, touch, and kinesthetics. From these short stories and reading sessions emerged Redwall, an 800-page handwritten manuscript.

Redwall
During his time at the Everyman Theatre, Jacques had met and become friends with Alan Durband, an English teacher at C. F. Mott Training College, a writer, and co-founder of the Everyman. Wanting Durband's opinion of Redwall, Jacques gave him the completed manuscript. Impressed, Durband then showed it to his own publisher without telling Jacques. Durband reportedly told his publishers: "This is the finest children's tale I've ever read, and you'd be foolish not to publish it"; Jacques was summoned to London to meet with the publishers, who gave him a contract to write the next five books in the series.

Redwall was unusual for its length. Although it is now common for children's books to have 350 pages, and the Harry Potter books far exceed that, at the time it was commonly regarded that 200 pages were the maximum that would hold a child's attention. It set the tone for the series as a whole, centering on the triumph of good over evil, with peaceful mice, badgers, voles, hares, moles and squirrels defeating rats, weasels, ferrets, snakes and stoats. Jacques did not shy away from the reality of battle, and many of the "good" creatures die.

Redwall alludes to the surrounding human civilization, for example with a scene featuring a horse-drawn cart. The subsequent books ignore humans completely, portraying an Iron Age society from the misty past building castles, bridges and ships to the scale of forest creatures, writing their own literature and drawing their own maps. Jacques was highly involved in the audio books of his work, even personally enlisting his sons and others to voice Redwall inhabitants. Jacques said that the characters in his stories are based on people he encountered in his life. He based Gonff, the self-proclaimed "Prince of Mousethieves", on himself when he was a young boy hanging around the docks of Liverpool. Mariel is based on his granddaughter. Constance the Badgermum is based on his maternal grandmother. Other characters are a combination of many of the people he had met in his travels.

Jacques lived through the rationing during and after the Second World War, when he fantasized about the dishes in his aunt's illustrated Victorian cookbook. Groaning boards spread with sumptuous feasts are common scenes in his stories, described in intricate sensory detail. The war also informed his depictions of gruesome battles. Jacques was known to be old-fashioned in his living; he thought an old typewriter to be more reliable than a computer, and he was known to be not fond of video games and other modern technology, though he allowed an animated television series to be produced based on his work, which aired on PBS in the United States. In the series, he introduced himself at the beginning of each episode and answered children's questions at the end, though the UK and Canadian airings omitted the Q&A session. He never felt that he fit the image of a "writer sitting in his garden." Nevertheless, he was deeply touched by his success at reaching children. He was also pleased to be recognized by the people of Liverpool. His novels have sold more than 20 million copies worldwide and have been published in 28 languages.

Death
In 2011, Jacques was admitted to the Royal Liverpool Hospital to undergo emergency surgery for an aortic aneurysm. He died from a heart attack at 71 years old on 5 February 2011.

Other interests
Jacques also had musical interests. In the 1960s he formed a folk music band, the Liverpool Fishermen, with two of his brothers. He hosted a radio show called Jakestown on BBC Radio Merseyside from 1986 to 2006, featuring selections from his favourite operas.

Personal life
Jacques lived with his wife, Maureen, in Liverpool. They had two sons: Marc is a carpenter and bricklayer, and David a contemporary artist and muralist.

Recognition
In June 2005, he was awarded an Honorary Doctorate of Letters by the University of Liverpool. A prize was created at Bristol Grammar School, known as the 'Brian Jacques Award for Most Improved Creative Writing', and is awarded to a student in Year 8 as book tokens.

Redwall series

 * 1) Redwall (1986)
 * 2) Mossflower (1988)
 * 3) Mattimeo (1989)
 * 4) Mariel of Redwall (1991)
 * 5) Salamandastron (1992)
 * 6) Martin the Warrior (1993)
 * 7) The Bellmaker (1994)
 * 8) Outcast of Redwall (1995)
 * 9) The Pearls of Lutra (1996)
 * 10) The Long Patrol (1997)
 * 11) Marlfox (1998)
 * 12) The Legend of Luke (1999)
 * 13) Lord Brocktree (2000)
 * 14) The Taggerung (2001)
 * 15) Triss (2002)
 * 16) Loamhedge (2003)
 * 17) Rakkety Tam (2004)
 * 18) High Rhulain (2005)
 * 19) Eulalia! (2007)
 * 20) Doomwyte (2008)
 * 21) The Sable Quean (2010)
 * 22) The Rogue Crew (2011) (posthumous)

Tribes of Redwall series

 * Tribes of Redwall Badgers (2001)
 * Tribes of Redwall Otters (2001)
 * Tribes of Redwall Mice (2003)
 * Tribes of Redwall Squirrels (Unreleased)
 * Tribes of Redwall Hares (Unreleased)

Miscellaneous Redwall books

 * The Great Redwall Feast (1996)
 * Redwall Map & Riddler (1997)
 * Redwall Friend & Foe (2000)
 * A Redwall Winter's Tale (2003)
 * The Redwall Cookbook (2005)

Castaways of the Flying Dutchman series

 * Castaways of the Flying Dutchman (2001)
 * The Angel's Command (2003)
 * Voyage of Slaves (2006)

Urso Brunov

 * The Tale of Urso Brunov: Little Father of All Bears (2003)
 * Urso Brunov and the White Emperor (2008)

Other works

 * Seven Strange and Ghostly Tales (1991)
 * The Ribbajack & Other Curious Yarns (2004)
 * Get Yer Wack (Anvil Press, 1971)
 * YENNOWORRAMEANLIKE (Raven Books, 1972)
 * According to Jacques - A Mersey Bible (Raven Books, 1975)
 * Jakestown, My Liverpool (Raven Books, 1979)

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Talk:Carl Benjamin Boyer

Notability
Moved from Talk:Daniel C. Boyer (writen by User:Wwheeler) Hey! I have read "A History of Mathematics". It's great!

Keep that "He was the grandfather of Daniel C. Boyer" garbage out of there. This guy was basically Non-notable and this article should probably be AfD'd, it's even worse to have this guy's grandson, who is totally non-noteable, insist on putting his own name in the article. Now if you don't mind, I'm off to edit Cyrus McCormick to say "He was the grandfather of the cousin of the uncle of Descendall" --Descendall 05:28, 3 March 2006 (UTC)

I do not agree that he was non-notable. Columbia University has an award in his name, and he did make contribute to the history of mathematics (A History of Mathematics is a useful source for anyone writing about it here). P.L.A.R. 22:23, 18 July 2006 (UTC)

I can't believe anyone would say such a thing of Carl B. Boyer. Who cares about his grandson? I read his book called "The History of calculus and its conceptual development". In my opinion, anyone who calls himself a professor of mathematics without having read this book is a fraud! Boyer was an original thinker and a man of profound intelligence. None of the Wiki puppets and fools who run this site are in any position to criticize his work. 66.60.241.7 (talk)

I also want to agree with the above comment regarding notability. Boyer was one of the foremost historians of mathematics in the twentieth century, if not the foremost. His book on the history of calculus is a classic work, which should be read by every mathematics major. It's amusing that anyone would suggest removing this article because Boyer isn't notable enough when Wikipedia is filled with articles about thousands of two-bit rock bands that nobody cares about. Daniel Grünfeld (talk) 22:52, 20 March 2009 (UTC)

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title: Talk:Carl Benjamin Boyer


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Culture of Liberia

The culture of Liberia reflects this nation's diverse ethnicities and long history. Liberia is located in West Africa on the Atlantic Coast.

Languages
The official language of Liberia is English. There are also more than 16 indigenous languages. Among the most widely studied Liberian languages in schools and universities are Kpelle and Bassa languages and to a lesser extent, Vai. Loma and Mende also have their own unique alphabets but are studied less.

Both languages are noted for their unique alphabets and phonetics that are not based on the Latin alphabet, or any European language but emerged from visions of each language's inventor. Bassa alphabet was popularized by Dr. Thomas Narvin Lewis in the early 20th century, after attending studies in the U.S at Syracuse University. He modeled it after he came into contact with former slaves of Bassa origin in Brazil and the West Indies who were still using the alphabet. Vai is another well known ancient script from Liberia, but distinct from the Bassa alphabet.

Among the listed alphabets, Bassa Vah is compared to include Armenian, Coptic (used by Egyptian Coptic and Coptic Church), Avestan used in Ancient Persia to write sacred hymns of Zoroastrianism, Georgian language in Republic of Georgia, Mongolian, Meroitic alphabet of ancient Sudan and parts of Nile Valley, and many other ancient scripts, Greek-based and Cyrillic alphabets. Some Vah letters resemble certain letters from the Ge'ez alphabet of Ethiopia and Eritrea, N'ko alphabet in Guinea, and Armenian alphabet.

The syllabary of Vai also been compared to Hiragana and Katakana scripts also syllabary for Japanese language. However many linguists hold that Bassa and Vai alphabets are unique in their own right for being African languages with alphabets alongside Ge'ez, N'ko, and Tifinagh (used to write Berber languages) that are neither influenced by Latin or Western writing system.

Religion
According To the 2008 National Census, 85.5% of Liberia's population practices Christianity. Muslims comprise 12.2% of the population, largely coming from the Mandingo and Vai ethnic groups.

Belief in traditional supernatural world
Regardless of public statements of identification with Christianity, a "vast majority" of Liberians believe a supernatural world of ancestral and bush spirits that impact daily life. Ethnic groups in all regions of Liberia participate in the traditional religious practices of the Poro and Sande secret societies, with the exception of the Krahn ethnic group, who have their own secret society.

"Liberian religious culture is characterised by a predisposition towards secrecy (encapsulated in the concept of ifa mo - "do not speak it") and an ingrained belief in the intervention of mysterious forces in human affairs". "Both elite and non-elite Liberians usually attribute events to the activities of secret powers and forces".

"Beliefs include the conviction that there are deep and hidden things about an individual that only diviners, priests, and other qualified persons can unravel. This presupposes that whatever exists or happens in the physical realm has foundations in the spirit world".

Secret societies
While still belonging to Christian or Muslim denominations, ethnic groups in all regions participate in religious secret societies associated with the spirit world, including the Poro and Sande gender-based secret societies. The Sande society champions women's social and political interests and promotes their solidarity vis-a-vis the Poro society, a complementary secret society for men. The Poro society is found among the Bassa, Gola, Kissi, Kpelle, Loma, People and Vai of Liberia.

Polygamy
One-third of married Liberian women between the ages of 15–49 are in polygamous marriages. Customary law allows men to have up to four wives.

Media culture
Numerous newspapers, radio stations and TV programs are broadcast and can be heard in the capital Monrovia, coastal cities and towns and countryside. Radio, newspapers and online news articles are the main form of mass communication in Liberia in recent years, surpassing TV stations as the most accessible forms of media to Liberians. Many popular FM radio stations have their headquarters in Monrovia along with several major national newspapers.

Many radio stations are community-based, operated by joint United Nations and community councils, activists, youth groups, universities and neighborhood programs. The major radio stations in Liberia are UNMIL Radio, Radio ELWA, Truth Radio, ELBC Radio, and STAR radio. All have programs available to listen to online. There are currently no AM radio stations (which existed before the war) but there are a few shortwave stations. Radio also serves to promote peace, reconciliation and connect the country both rural and urban Liberians through community based apprenticeship programs for youths and young adults.

A famous news blackboard in Monrovia is The Daily Talk. The Daily Talk is accessible for free. According to The New York Times, the blackboard is "the most widely read report" in Monrovia, as many Monrovians lack the money or the electricity necessary for access to the conventional mass media.

The most widely read newspapers include the Liberia Herald, The Analyst, Liberian Observer, The News, The Heritage, and The Inquirer, among others. Several Liberian journalists have been awarded national and international awards and acclaim around the world for their commitment to press freedom and promotion of Democracy in post war countries and regions.

Music
Liberia has its own ancient music and instruments. While Liberian music is part of wider West African music heritage, it is also distinct from its neighbors. There are several different types of drums used in traditional music. Drums are among the most widely used instruments in ceremonies both official and nonofficial, weddings, christenings, naming ceremonies, holidays, graduations, etc. Next to drums, beaded gourd rattles called Saasaa are also used in mainstream music by many Liberian singers, musicians, and ensembles across the country.

Songs are sung in both English and all indigenous languages. Other instruments similar to the xylophone include Yomo Gor. Music is a main highlight of Liberian culture not only used as entertainment but to educate society on issues ranging from culture, politics, history to human rights.

Religious music is also popular. Christian music is heavily influenced by its counterpart in the United States, regardless of region. Islamic nasheeds popular in many countries with Muslim communities are almost unheard of in Liberia. Instead, music for Liberian Muslims is based on Quranic citations, adhan and music related to everyday life called suku. Aside from religious and traditional music, rap and Highlife are widely popular genres, especially with younger Liberians and American music afficionados, and can be heard in discothèques, parties, clubs and on radios throughout the country. Jazz, funk, soul, rap and a new music style or Liberian rap called Hipco combining rap, R&B, traditional rhymes, and joint Liberian and American influences are part of the wider music of Liberia.

Crafts
Liberia is renowned for its detailed decorative and ornate masks, large and miniature wood carvings of realistic human faces, famous people, scenes of everyday life, and accessories particularly combs, spoons, and forks which are often enlarged sculptures. Sculptures are produced in both the countryside and cities. Liberian wood curved sculptures are heavily influenced by ancient history predating modern Liberia, folklore, proverbs, spirituality, rural life and show the artist's strong observations for grand detail and their connections to the people and objects sculpted. Liberian artists both in the country and diaspora have also gained recognition for various styles of paintings in abstract, perspective, and graphic art.

Due to its strong relationship with the United States, Liberia has also produced its own American-influenced quilts. The American free people of color and former slaves who emigrated to Liberia brought with them their sewing and quilting skills and was originally done by Americo-Liberians beginning in the 19th century. Queen Victoria invited Martha Ann Ricks, on behalf of Liberian Ambassador Edward Wilmot Blyden, to Windsor Castle on 16 July 1892.

Ricks, a former slave from Tennessee, had saved for more than fifty years, to afford the voyage from Liberia to England to personally thank the Queen for the British navy's actions against the slave trade. Ricks shook hands with the Queen and presented her with a Coffee Tree quilt, which Victoria later sent to the 1893 World's Columbian Exposition for display. A mystery remains as to where the quilt is today.

The census of 1843 indicated a variety of occupations, including hatter, milliner, seamstress, and tailor. Liberia hosted National Fairs in 1857 and 1858 in which prizes were awarded for various needle arts. However, today, Liberians from all ethnic groups make quilts, although quilting is not as popular as it once was when the country was adapting American customs, culture, and lifestyle in the mid-19th century. In modern times, Liberian presidents would present quilts as official government gifts, and when current Liberian President Ellen Johnson Sirleaf moved into the Executive Mansion, she reportedly had a Liberian-made quilt installed in her presidential office.

Literature
A literary tradition has existed in Liberia for over a century. Liberia had no written tradition until the 19th century. Numerous Liberian authors throughout the years have contributed to the writings of various genres. They have written on folk art, ancient proverbs, everyday life in countryside, city life, religion and observation of their own lives. Culture, tradition, identity, society, taboo subjects, human rights, equality and diversity within Liberia, multiculturalism, Pan-Africanism, colonialism and its reverberating consequences today, post-colonial African countries and future of the country have been featured in novels, books, magazines, and novelettes since the 19th century. The first novel by an African published in English was written by a Liberian, Joseph Jeffrey Walters, a Vai convert to Christianity studying at Oberlin College in the United States. The book titled Guanya Pau: A Story of an African Princess was published in 1891 by Lauer and Mattil of Cleveland, Ohio, and deals with the issue of child marriage and women's rights.

Poetry is a prominent canon of Liberian literature. Many authors have presented their pose in all poetic styles. Often adding their own unique perspectives, writing styles and observation of the material and spiritual worlds into their books. Liberia's prominent writers also share a variety of genres that cross several decades. In the 19th century, Edward Wilmot Blyden was the most renowned Liberian author. A diplomat, educator, statesman and writer, Blyden was considered one of the early fathers of Pan Africanism along with W. E. B. Du Bois and Marcus Garvey. His writings revolved around the need for Africans to develop their own identity, be culturally, spiritually and politically aware of their own potential and preside over their own self-rule and to disprove the European view of Africans as culture-less. These writings inspired many Liberian authors in later years and still do today. A national hero in Liberia, Blyden is known in world history and on the African continent as the genius behind the phrase "Africa for Africans!" and later inspired Marcus Garvey's Back to Africa Movement.

During the 20th century and in the first decade of the 21st century, other authors have taken a less political but prominent role in Liberian literature. Several authors are renowned for their detailed and deep observation of Liberian life both in the country and abroad in the Diaspora in Europe or United States. Authors Bai T. Moore, E. G. Bailey, Roland T. Dempster, Wilton G. S. Sankawulo, have all reflected on Liberian culture, tradition, modernization and the pain of exile, loneliness, loss and remembrance in fiction, and non-fiction works.

Bai T. Moore's novella Murder in the Cassava Patch is required reading for many Liberian high-school students. Published in 1968, the book is based on the true story of a murder, and explores the most taboo subjects in mid-20th-century Liberia as the story reveals the lives of the main characters and their hometown. E. G. Bailey is a spoken-word artist, theatre and radio producer.

Politician and author Wilton G. S. Sankawulo published many collections of poems and stories that later became praised anthologies on Liberian folklore and wider African literary tradition, entitled More Modern African Stories. His most celebrated book is Sundown at Dawn: A Liberian Odyssey. According to the book's publisher, Dusty Spark Publishing, it is regarded as "one of the literary achievements of postwar Liberia and contemporary Africa."

Cuisine
Liberian cuisine has rice as its staple. Other ingredients include cassava, fish, bananas, citrus fruit, plantains, coconut, okra and sweet potatoes. Heavy stews spiced with habanero and scotch bonnet chillies are popular and eaten with fufu.

title: Culture of Liberia


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Garth Nix

Garth Richard Nix (born 19 July 1963) is an Australian writer who specialises in children's and young adult fantasy novels, notably the Old Kingdom, Seventh Tower and Keys to the Kingdom series. He has frequently been asked if his name is a pseudonym, to which he has responded, "I guess people ask me because it sounds like the perfect name for a writer of fantasy. However, it is my real name."

Biography
Born in Melbourne, Nix was raised in Canberra. He attended Turner Primary School, Lyneham High School and Dickson College for schooling. While at Dickson College, Nix joined the Australian Army Reserve. After a period working for the Australian government, he traveled in Europe before returning to Australia in 1983 and undertaking a BA in professional writing at University of Canberra.

He worked in a Canberra bookshop after graduation, before moving to Sydney in 1987, where he worked his way up in the publishing field. He was a sales rep and publicist before becoming a senior editor at HarperCollins. In 1993 he commenced further travel in Asia, the Middle East and Eastern Europe before becoming a marketing consultant, founding his own company, Gotley Nix Evans Pty Ltd. From 1999 to 2002 he worked as a literary agent with Curtis Brown (Australia) Pty Ltd before becoming a full-time author.

In addition to his work as a fantasy novelist, Nix has written a number of scenarios and articles for the role playing field, including those for Dungeons & Dragons and Traveller. These have appeared in related publications such as White Dwarf, Multiverse and Breakout!. He has also written case studies, articles and news items in the information technology field, his work appearing in publications such as Computerworld and PCWorld.

Nix currently lives in Sydney with his wife Anna McFarlane, a publisher, and their sons Thomas and Edward.

Very Clever Baby
This series was self-published, and republished by Text Publishing. Described as books for "Very Clever Babies Aged 3–6 Months", they contain such words as ichthyologist, as used by the character Freddy the Fish.
 * Very Clever Baby's First Reader (1988)
 * Very Clever Baby's Ben Hur (1988)
 * Very Clever Baby's Guide to the Greenhouse Effect (1992)
 * Very Clever Baby's First Christmas (1998)

Novels

 * Sabriel (1995)
 * Lirael (2001)
 * Abhorsen (2003)
 * Clariel (2014)
 * Goldenhand (2016)
 * Terciel and Elinor (2021)

Short fiction

 * "Nicholas Sayre and the Creature in the Case" (2005). A novella produced for World Book Day and originally entitled "The Creature in the Cave". Reprinted under the new title in the collection Across the Wall: A Tale of the Abhorsen and Other Stories.)
 * "The Nine Gates of Death: An Extract of the Journal of Idrach the Lesser Necromancer" (2009). A short story released on oldkingdom.com.au
 * "An Essay on Free Magic". Short text released on the Old Kingdom website.
 * "To Hold the Bridge". A novella original published in the Legends of Australian Fantasy anthology edited by Jack Dann and Jonathan Strahan. Reprinted in the collection To Hold the Bridge.
 * "Doctor Crake Crosses the Wall". Short story included in the Australian edition of Goldenhand, also published on the official Australian Old Kingdom website.

Omnibus

 * The Old Kingdom Chronicles (The Abhorsen Chronicles in the United States) (2009). This contained the first three Abhorsen books and "The Creature in the Case".

The Seventh Tower

 * 1) The Fall (2000)
 * 2) Castle (2000)
 * 3) Aenir (2001)
 * 4) Above the Veil (2001)
 * 5) Into Battle (2001)
 * 6) The Violet Keystone (2001)

The Keys to the Kingdom

 * 1) Mister Monday (2003)
 * 2) Grim Tuesday (2004)
 * 3) Drowned Wednesday (2005)
 * 4) Sir Thursday (2006)
 * 5) Lady Friday (2007)
 * 6) Superior Saturday (2008)
 * 7) Lord Sunday (2010)

The Left-Handed Booksellers of London

 * 1) The Left-Handed Booksellers of London (2020)
 * 2) The Sinister Booksellers of Bath (2023)

Troubletwisters (co-written with Sean Williams)

 * 1) Troubletwisters (2011)
 * 2) The Monster (2012)
 * 3) The Mystery (2013)
 * 4) The Missing   (2014)

Have Sword, Will Travel (co-written with Sean Williams)

 * 1) Have Sword, Will Travel (2017)
 * 2) Let Sleeping Dragons Lie (2018)

Works in multi-author series

 * Spirit Animals Book 3: Blood Ties (2014, with Sean Williams)
 * Serena and the Sea Serpent (2000), part of Aussie Bites series
 * The Calusari (1997), an adaptation of the X-Files episode of the same name

Standalone novels

 * The Ragwitch (1990), a children's fantasy novel
 * Shade's Children (1997)
 * A Confusion of Princes (2012), a young adult space opera novel
 * Newt's Emerald (2015), a fantasy romance
 * Frogkisser! (2017), a children's fantasy novel
 * Angel Mage (2019), a young adult fantasy sci-fi novel

Sir Hereward and Mister Fitz

 * 1) 2007 "Sir Hereward and Mister Fitz Go to War Again", in Jim Baen's Universe
 * 2) 2008 "Beyond the Sea Gate of the Scholar-Pirates of Sarsköe", in Fast Ships, Black Sails edited by Jeff VanderMeer and Ann VanderMeer
 * 3) 2010 "A Suitable Present for a Sorcerous Puppet", in Swords and Dark Magic, edited by Lou Anders and Jonathan Strahan
 * 4) 2013 "Losing Her Divinity", in Rags & Bones: New Twists on Timeless Tales, edited by Melissa Marr and Tim Pratt
 * 5) 2014 "A Cargo of Ivories", in Rogues, edited by George R. R. Martin and Gardner Dozois
 * 6) 2014 "Home is the Haunter", in Fearsome Magics, edited by Jonathan Strahan
 * 7) 2017 "A Long, Cold Trail", in The Book of Swords, edited by Gardner Dozois
 * 8) 2020 "Cut Me Another Quill, Mister Fitz", in The Book of Dragons, edited by Jonathan Strahan
 * 9) 2023 "The Field of Fallen Foe", in the collection Sir Hereward and Mister Fitz: Stories of the Witch Knight and the Puppet Sorcerer (2023)
 * The first three stories are collected in Sir Hereward and Mister Fitz: Three Adventures (2011).
 * All eight stories plus a new ninth story The Field of Fallen Foe are collected in Sir Hereward and Mister Fitz: Stories of the Witch Knight and the Puppet Sorcerer (2023).

Collections

 * 2007 One Beastly Beast: Two Aliens, Three Inventors, Four Fantastic Tales - a book of short stories for younger readers
 * "Serena and the Sea Serpent"
 * "Bill the Inventor"
 * "Blackbread the Pirate"
 * "The Princess and the Beastly Beast"
 * 2005 Across the Wall: A Tale of the Abhorsen and Other Stories
 * 2005 "Nicholas Sayre and the Creature in the Case" (published for World Book Day)
 * 2001 "Under the Lake" (from The Magazine of Fantasy & Science Fiction)
 * 2005 "Charlie Rabbit" (from Kids' Night In collected for War Child)
 * 1996 "From the Lighthouse" (from Fantastic Worlds anthology edited by Paul Collins)
 * 2001 "The Hill" (from X-Changes: Stories for a New Century)
 * 2001 "Lightning Bringer" (from Love & Sex anthology edited by Michael Cart)
 * 1987 "Down to the Scum Quarter" (fromMyths and Legends, reprinted inBreakout! magazine in 1988)
 * 2002 "Heart's Desire" (from The Magazine of Fantasy & Science Fiction)
 * 2000 "Hansel's Eyes" (from A Wolf at the Door anthology edited by Ellen Datlow and Terri Windling)
 * 2003 "Hope Chest" (from Firebirds anthology edited by Sharyn November)
 * 1999 "My New Really Epic Fantasy Series" (from Swancon Program Book)
 * 2000 "Three Roses" (from Eidolon magazine, Autumn 2000)
 * 2004 "Endings" (from Gothic! Ten Original Dark Tales anthology edited by Deborah Noyes)
 * 2015 To Hold the Bridge
 * 2010 "To Hold the Bridge: An Old Kingdom Story" (fromLegends of Australian Fantasy anthology, edited by Jack Dann and Jonathan Strahan)
 * 2011 "Vampire Weather" (from Teeth anthology, edited by Ellen Datlow and Terri Windling)
 * 2008 "Strange Fishing in the Western Highlands" (from Hellboy: Oddest Jobs, edited by Christopher Golden_
 * 2008 "Old Friends" (from Dreaming Again anthology, edited by Jack Dann)
 * 2011 "The Quiet Knight" (from Geektastic anthology edited by Holly Black and Cecil Castellucci)
 * 2012 "You Won't Feel a Thing" (from After anthology edited by Ellen Datlow and Terri Windling)
 * 2012 "A Handful of Ashes" (from Under My Hat anthology, edited by Jonathan Strahan)
 * 2012 "The Big Question" (from Elsewhere, Edinburgh Festival Special)
 * 2009 "Stop!" (from The Dragon Book anthology edited by Jack Dann and Gardner Dozois)
 * 2008 "Infestation" (from The Starry Rift anthology edited by Jonathan Strahan)
 * 2011 "The Heart of the City" (from Subterranean Online magazine)
 * "Ambrose and the Ancient Spirits of East and West" (from The Thackery T. Lambshead Cabinet of Curiosities anthology edited by Ann VanderMeer and Jeff VanderMeer)
 * 2007 "Holly and Iron" (from Wizards anthology, edited by Jack Dann and Gardner Dozois)
 * 2011 "The Curious Case of the Moondawn Daffodils Murder: As Experienced by Sir Magnus Holmes and Almost-Doctor Susan Shrike" (from Ghosts by Gaslight anthology edited by Jack Dann and Nick Gevers)
 * 2009 "An Unwelcome Guest" (from Troll's-Eye View anthology edited by Ellen Datlow and Terri Windling)
 * 2010 "The Highest Justice" (from Zombies vs. Unicorns anthology edited by Holly Black and Justine Larbalestier)
 * 2012 "Master Haddad's Holiday" (a bonus story from the Australian printing of A Confusion of Princes)
 * 2012 "Sidekick of Mars" (originally from the Under the Moons of Mars anthology edited by John Joseph Adams)
 * 2011 "Peace in Our Time" (originally from the Steampunk! anthology edited by Kelly Link and Gavin J. Grant)
 * 2011 Sir Hereward and Mister Fitz: Three Adventures
 * 2007 "Sir Hereward and Mister Fitz Go To War Again" (from Jim Baen's Universe)
 * 2008 "Beyond the Sea Gate of the Scholar-Pirates of Sarsköe" (from Fast Ships, Black Sails anthology edited by Jeff and Ann VanderMeer)
 * 2010 "'A Suitable Present for a Sorcerous Puppet'" (from Swords and Dark Magic anthology edited by Lou Anders and Jonathan Strahan)

Uncollected stories

 * 1984 "Sam, Cars and the Cuckoo" in Warlock magazine no. 2
 * 1996 "The Kind Old Sun Will Know" first published in Eidolon magazine
 * 2005 "Read It in the Headlines!" in Daikaiju! Giant Monster Tales, edited by Robert Hood and Robin Pen
 * 2006 "Dog Soldier" first published in Jim Baen's Universe, 2006
 * 2007 "Bad Luck, Trouble, Death and Vampire Sex" first published in Eclipse, edited by Jonathan Strahan
 * 2009 "The Nine Gates of Death: An Extract of the Journal of Idrach the Lesser Necromancer" – first published on oldkingdom.com.au
 * 2010 "The Highest Justice" in Zombies vs. Unicorns, edited by Justine Larbalestier and Holly Black
 * 2013 "Crossing the Line" first published in Fearie Tales, edited by Stephen Jones
 * 2013 "Fire Above, Fire Below" first published by Tor.com
 * 2014 "Shay Corsham Worsted" first published in Fearful Symmetries, edited by Ellen Datlow
 * 2014 "Happy Go Lucky" first published in Kaleidoscope, edited by Alisa Krasnostein and Julia Rios
 * 2015 "By Frogsled and Lizardback to Outcast Venusian Lepers" in Old Venus, edited by George R. R. Martin and Gardner Dozois
 * 2019 "Dislocation Space" first published by Tor.com
 * 2020 "The Case of the Somewhat Mythic Sword" first published by Tor.com
 * 2020 "The Necessary Arthur" first published by Tor.com

title: Garth Nix


text:
Carol Berg

Carol Berg (born 1948) is the author of fantasy novels, including the books from the Rai-Kirah series, Song of the Beast, the books from The Bridge of D'Arnath series, the Lighthouse novels, and Collegia Magica. She also writes the Chimera series under the pen name Cate Glass.

Berg holds a degree in mathematics from Rice University, and a degree in computer science from the University of Colorado. Before writing full-time, she designed software. She lives in Colorado, and is the mother of three boys.

The Rai-Kirah series

 * 1) Transformation (ISBN 0-451-45795-1) (2000)
 * 2) Revelation (ISBN 0-451-45842-7) (2001)
 * 3) Restoration (ISBN 0-451-45890-7) (2002)

The Bridge of D'Arnath series

 * 1) Son of Avonar (ISBN 0-451-45962-8) (2004)
 * 2) Guardians of the Keep (ISBN 0-451-46000-6) (2004)
 * 3) The Soul Weaver (ISBN 0-451-46017-0) (2005)
 * 4) Daughter of Ancients (ISBN 0-451-46042-1) (2005)

The Lighthouse Duet

 * 1) Flesh and Spirit (ISBN 0-451-46088-X) (2007)
 * 2) Breath and Bone (ISBN 0-451-46186-X) (2008), the winner of the 2008 Colorado Book Award and the 2009 Mythopoeic Fantasy Award

The Sanctuary Duet
In the same setting as The Lighthouse Duet.

 * 1) Dust and Light (ISBN 0-451-41724-0) (ISBN 978-0451417244) (2014)
 * 2) Ash and Silver (ISBN 0-451-41726-7) (ISBN 978-0451417268) (2015)

Novels of the Collegia Magica

 * 1) The Spirit Lens (ISBN 0-451-46311-0) (2010)
 * 2) The Soul Mirror (ISBN 0-451-46374-9) (2011), the winner of the 2011 Colorado Book Award
 * 3) The Daemon Prism (ISBN 0-451-46434-6) (2012)

The Chimera series (as Cate Glass)

 * 1) An Illusion of Thieves (ISBN 9781250311009) (2019)
 * 2) A Conjuring of Assassins (ISBN 9781250311023) (2020)
 * 3) A Summoning of Demons (ISBN 9781250311054) (2021)

Other works

 * 1) Song of the Beast, fantasy standalone (ISBN 0-451-45923-7) (2003) and winner of the 2003 Colorado Book Award
 * 2) "Unmasking", a novella set in the world of the Rai-kirah in the collection Elemental Magic (ISBN 978-0425217863) (2007)
 * 3) "At Fenwick Faire", a gritty myth; in Broken Links, Mended Lives, RMFW Press (ISBN 978-0976022527) (2009)
 * 4) "Seeds", a short story set in the kingdom of Navronne; in Blackguards: Tales of Assassins, Mercenaries and Rogues, Ragnarok Publications (ISBN 978-1941987063) (2015)
 * 5) "The Heart's Coda," novelette follow-up to Song of the Beast; in Lace and Blade 4, Marion Zimmer Bradley Literary Trust (ISBN 9781938185519) (2018)

Awards

 * Finalist, (2001) Compton Crook Award, Transformation
 * Finalist, (2000) Barnes & Noble Maiden Voyage Award, Transformation
 * Finalist, (2002) Romantic Times Reviewers' Choice Award for Best Epic Fantasy, Restoration
 * Winner, (2003) Colorado Book Award, Song of the Beast
 * Winner, (2005) Geffen Award for Best Translated Fantasy, Transformation
 * Finalist, (2006) RWA-FFP Prism Award for Best Romantic Fantasy, Daughter of Ancients
 * Finalist, (2006) Colorado Book Award, Daughter of Ancients
 * Finalist, (2008) Colorado Book Award, Flesh and Spirit
 * Winner, (2009) Colorado Book Award, Breath and Bone
 * Winner, (2009) Mythopoeic Fantasy Award for Adult Literature, Flesh and Spirit,Breath and Bone
 * Finalist, (2010) Colorado Book Award, The Spirit Lens
 * Winner, (2011) Colorado Book Award, The Soul Mirror

title: Carol Berg


text:
Odense River

The Odense River (Odense Å) is a river located on the island of Funen, in central Denmark. It is about 60 km long and is named after the Funish capital, Odense, which it passes through. Boats can be rented by the river, offering a scenic ride to Fruens Bøge. Excursion boats offer rides to Carlslund, with jazz music some Saturdays during the Summer.

During the Viking Age, the fortress Nonnebakken ensured its controller supremacy over the river.

title: Odense River


text:
Zoe Ball

Zoe Louise Ball (born 23 November 1970) is a British broadcaster and presenter. She was the first female host of the Radio 1 and Radio 2 breakfast shows for the BBC, and presented the 1990s children's show Live & Kicking, alongside Jamie Theakston from 1996–1999.

Ball was a contestant in the third series of Strictly Come Dancing. Following this, in 2011 she replaced Claudia Winkleman as host of the BBC Two spin-off show Strictly Come Dancing: It Takes Two until her departure in 2021. Ball also hosted the Strictly Come Dancing Live Tour in 2011 and 2015.

In 2018, Ball was announced as the new host of BBC Radio 2's flagship breakfast show and took over from Chris Evans in January 2019.

Early life
Zoe Ball was born in Blackpool, Lancashire, and grew up in Farnham Common, Buckinghamshire. She is the daughter of the children's TV presenter Johnny Ball and his wife Julia (née Anderson). The couple divorced when Zoe was two, Zoe was raised with her father.

Ball was educated at Heston Junior School in the district of Hounslow between 1975 and 1978. Her family then moved to Farnham Common in Buckinghamshire. She attended Farnham Common First School and Farnham Common Middle School before moving to Holy Cross Convent School in Chalfont St Peter and Amersham College of Art and Technology for a "City & Guilds in radio and journalism".

She also attended the Young Theatre at Beaconsfield where she trained as an actress. Her first TV appearance was as a child in the studio audience of the 1980s Saturday morning children's show, Saturday Superstore, on which her father was appearing as a guest.

Television
Ball began her television career as a presenter on the pre-school programme Playdays.

She became a runner at Granada Television and researcher for Cool Cube (1990–1994), a children's Saturday afternoon magazine show, lensed at Granada's studio in Manchester, produced by Clear Idea Television, and broadcast on the on British Satellite Broadcasting channel Galaxy.

In 1994-1996, Ball presented SMart with Mark Speight and Jay Burridge. In 1995, Ball hosted Fully Booked for the first series. Between 1996 and 1998, she was a regular presenter on Top of the Pops, usually alternating with fellow presenters and DJs Jayne Middlemiss and Jo Whiley.

In 1996, she was a co-host with Jamie Theakston in BBC One's Saturday morning children's program Live & Kicking. In 1998, television personality Chris Evans was a guest of the show, and later employed both Ball and Theakston to co-host the Channel 4 Wednesday night chat/music show The Priory, under Ginger Productions.

Her presenting jobs have also included hosting The Big Breakfast on Channel 4,

Ball co-hosted the 2002 Brit Awards with Frank Skinner, In 2005, she co-hosted the BBC reality show Strictly Dance Fever. In 2006, Ball co-hosted the ITV programme Extinct, alongside Trevor McDonald.

In January 2007, she presented the second series of ITV's Soapstar Superstar, taking over from Fern Britton and Ben Shephard, who hosted the first series. In March 2007, she hosted the ITV talent search Grease Is the Word.

Alongside Jamie Theakston, Ball presented Channel 5's quiz series Britain's Best Brain in 2009. Ball has guest presented several episodes of The One Show as a stand-in for Alex Jones.

On 4 August 2013, as hostess of a special entitled Doctor Who Live: The Next Doctor, she revealed Peter Capaldi as the incoming Twelfth Doctor and hosted his first interview in that capacity.

In November 2013, she co-hosted the BBC One Children in Need telethon, alongside Sir Terry Wogan, Fearne Cotton, Nick Grimshaw and Tess Daly. In 2014, Ball presented a BBC Two spin-off from The Voice UK called The Voice: Louder on Two. The show aired for one series of ten episodes.

In 2015, Ball hosted EastEnders: Backstage Live, a spin-off show during the EastEnders live week. She co-presented the BBC Young Dancer competition with Darcey Bussell on BBC Two.

She also narrated two ITV Specials, The Nation's Favourite '70s Number One and The Nation's Favourite '80s Number One as well as the BBC One series Don't Tell the Bride. In 2016, Ball co-presented Can't Touch This, a Saturday night game show for BBC One, alongside Ashley Banjo.

She narrated The Nation's Favourite Carpenters Song, and guest-presented an episode of Film 2016. In 2017, Ball co-presented The Big Family Cooking Showdown with Nadiya Hussain on BBC Two.

Since 2020, Ball has appeared alongside her son Woody Fred Cook on Celebrity Gogglebox.

In 2021, Ball appeared on The Masked Dancer, masked as Llama. She was the sixth celebrity to be unmasked.

In 2023, she appeared as herself in Episode 1 of the BBC One comedy Queen of Oz. Ball is seen and heard on her radio programme questioning the outrageous antics of spoiled spare to the British crown, Princess Georgiana, played by Catherine Tate.

In 2024, Ball appeared as herself on the "New Year's Treat" episode of Taskmaster.

Strictly Come Dancing and It Takes Two
In October 2005, Ball became a contestant on the third series of the BBC One talent show Strictly Come Dancing, where she was partnered by Ian Waite. Ball and Waite were ranked in third place; they also scored 38/40 (including two tens) for three dances in the series and one in the Christmas special.

In 2011, she took over from Claudia Winkleman as the host of Strictly Come Dancing's sister show It Takes Two, airing every weeknight on BBC Two; in 2014, whilst Winkleman was on leave after her daughter suffered serious burn injuries, Ball co-hosted the main show. On 17 May 2021, Ball announced she was leaving her role as presenter after 10 years.

Radio
Although known primarily for her TV work, Ball first worked in radio, after she became co-host of Radio 1 Breakfast on BBC Radio 1 in October 1997 with Kevin Greening. She became the sole host, the first female DJ to do so. At this time, her hard-drinking, hard-partying lifestyle contributed to the identification of the so-called "ladette culture" of the late 1990s. She was twice warned by the BBC for swearing on the radio.

Ball left BBC Radio 1 in March 2000 to start a family and a new chapter. Her final show was on 10 March 2000; she was succeeded by Sara Cox.

Ball returned to radio in mid-2002 when she joined London radio station Xfm, where she presented the weekday drivetime show until December 2003. In January 2004, she took over a Friday evening music show for the station. She also stood in for Ricky Gervais while he filmed the second series of The Office. She left Xfm at the end of 2004.

In September 2007, she returned to BBC Radio and co-hosted a show with Sara Cox, celebrating 40 years of BBC Radio 1 and BBC Radio 2.

From 2006, she provided relief presenting duties for BBC Radio 2, fronted specialist documentaries, sat in for Dermot O'Leary for three weeks in February 2006, and co-presented (with Danny Baker) the hastily conceived replacement for Jonathan Ross' Saturday morning show, in the wake of Ross's suspension due to Sachsgate in 2008.

In 2009, Ball became the usual relief presenter for Ken Bruce's weekday mid-morning show on BBC Radio 2. She also began hosting the Saturday breakfast show from 6:00 a.m. to 8:00 a.m. on the network from 6 June 2009 as part of a shake-up of weekend programming at Radio 2. Ball left Radio 2 for a while as her last show was broadcast on Saturday 28 January 2012. Her replacement in that slot was Anneka Rice.

Ball covered The Radio 2 Breakfast Show for Chris Evans on several occasions.

From 4 March 2017 until 22 December 2018, Ball returned to Radio 2 on permanent basis where she presented the Saturday afternoon slot on BBC Radio 2 between 3–6pm, taking over from Dermot O'Leary.

On 3 October 2018, she was announced by Chris Evans as the new host of The Radio 2 Breakfast Show, which began on 14 January 2019. Rylan Clark replaced Ball in the Saturday mid-afternoon slot.

Personal life
Whilst at BBC Radio 1, Ball began a relationship with the DJ and musician Norman Cook (also known as Fatboy Slim). The couple married at Babington House in Somerset in August 1999. In 2003, the couple split up when Ball revealed that she had had an affair; they later reconciled. The couple have a son, Woody Fred Cook (born 2000), and a daughter, Nelly (born 2010). On 24 September 2016, the couple announced their separation after 18 years together, and divorced in 2020.

In 2013 she started an affair with Billy Yates. He took his own life in May 2017 after a long battle with depression at the age of 40.

Between 2017 and 2023, Ball was in a relationship with fashion model and carpenter Michael Reed.

Ball celebrated two years of sobriety in 2018.

Charity work
In 2014, she supported Marks & Spencer and Oxfam's Love Mum campaign.

In March 2018, Ball took part in the 'Hardest Ride Home' challenge in which she cycled from Blackpool to Brighton to raise money for Sport Relief and mental health awareness. She set off on 5 March for five days. Highlights of the challenge were broadcast in a one-off documentary, airing on 21 March 2018 on BBC One. She raised £1,198,012.

title: Zoe Ball


text:
Seminorm

In mathematics, particularly in functional analysis, a seminorm is a norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm.

A topological vector space is locally convex if and only if its topology is induced by a family of seminorms.

Definition
Let $$X$$ be a vector space over either the real numbers $$\R$$ or the complex numbers $$\Complex.$$ A real-valued function $$p : X \to \R$$ is called a if it satisfies the following two conditions:

 * 1) Subadditivity/Triangle inequality: $$p(x + y) \leq p(x) + p(y)$$ for all $$x, y \in X.$$
 * 2) Absolute homogeneity: $$p(s x) =|s|p(x)$$ for all $$x \in X$$ and all scalars $$s.$$

These two conditions imply that $$p(0) = 0$$ and that every seminorm $$p$$ also has the following property: Nonnegativity: $$p(x) \geq 0$$ for all $$x \in X.$$

Some authors include non-negativity as part of the definition of "seminorm" (and also sometimes of "norm"), although this is not necessary since it follows from the other two properties.

By definition, a norm on $$X$$ is a seminorm that also separates points, meaning that it has the following additional property: Positive definite/Positive/: whenever $$x \in X$$ satisfies $$p(x) = 0,$$ then $$x = 0.$$

A is a pair $$(X, p)$$ consisting of a vector space $$X$$ and a seminorm $$p$$ on $$X.$$ If the seminorm $$p$$ is also a norm then the seminormed space $$(X, p)$$ is called a.

Since absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a sublinear function. A map $$p : X \to \R$$ is called a if it is subadditive and positive homogeneous. Unlike a seminorm, a sublinear function is necessarily nonnegative. Sublinear functions are often encountered in the context of the Hahn–Banach theorem. A real-valued function $$p : X \to \R$$ is a seminorm if and only if it is a sublinear and balanced function.

Examples
 The on $$X,$$ which refers to the constant $$0$$ map on $$X,$$ induces the indiscrete topology on $$X.$$ Let $$\mu$$ be a measure on a space $$\Omega$$. For an arbitrary constant $$c \geq 1$$, let $$X$$ be the set of all functions $$f: \Omega \rightarrow \mathbb{R}$$ for which $$\lVert f \rVert_c := \left( \int_{\Omega}| f |^c \, d\mu \right)^{1/c}$$ exists and is finite. It can be shown that $$X$$ is a vector space, and the functional $$\lVert \cdot \rVert_c$$ is a seminorm on $$X$$. However, it is not always a norm (e.g. if $$\Omega = \mathbb{R}$$ and $$\mu$$ is the Lebesgue measure) because $$\lVert h \rVert_c = 0$$ does not always imply $$h = 0$$. To make $$\lVert \cdot \rVert_c$$ a norm, quotient $$X$$ by the closed subspace of functions $$h$$ with $$\lVert h \rVert_c = 0$$. The resulting space, $$L^c(\mu)$$, has a norm induced by $$\lVert \cdot \rVert_c$$. If $$f$$ is any linear form on a vector space then its absolute value $$|f|,$$ defined by $$x \mapsto |f(x)|,$$ is a seminorm. A sublinear function $$f : X \to \R$$ on a real vector space $$X$$ is a seminorm if and only if it is a, meaning that $$f(-x) = f(x)$$ for all $$x \in X.$$ Every real-valued sublinear function $$f : X \to \R$$ on a real vector space $$X$$ induces a seminorm $$p : X \to \R$$ defined by $$p(x) := \max \{f(x), f(-x)\}.$$

Any finite sum of seminorms is a seminorm. The restriction of a seminorm (respectively, norm) to a vector subspace is once again a seminorm (respectively, norm).

If $$p : X \to \R$$ and $$q : Y \to \R$$ are seminorms (respectively, norms) on $$X$$ and $$Y$$ then the map $$r : X \times Y \to \R$$ defined by $$r(x, y) = p(x) + q(y)$$ is a seminorm (respectively, a norm) on $$X \times Y.$$ In particular, the maps on $$X \times Y$$ defined by $$(x, y) \mapsto p(x)$$ and $$(x, y) \mapsto q(y)$$ are both seminorms on $$X \times Y.$$

If $$p$$ and $$q$$ are seminorms on $$X$$ then so are $$(p \vee q)(x) = \max \{p(x), q(x)\}$$ and $$(p \wedge q)(x) := \inf \{p(y) + q(z) : x = y + z \text{ with } y, z \in X\}$$ where $$p \wedge q \leq p$$ and $$p \wedge q \leq q.$$

The space of seminorms on $$X$$ is generally not a distributive lattice with respect to the above operations. For example, over $$\R^2$$, $$p(x, y) := \max(|x|, |y|), q(x, y) := 2|x|, r(x, y) := 2|y| $$ are such that $$((p \vee q) \wedge (p \vee r)) (x, y) = \inf \{\max(2|x_1|, |y_1|) + \max(|x_2|, 2|y_2|) : x = x_1 + x_2 \text{ and } y = y_1 + y_2\}$$ while $$(p \vee q \wedge r) (x, y) := \max(|x|, |y|)$$

If $$L : X \to Y$$ is a linear map and $$q : Y \to \R$$ is a seminorm on $$Y,$$ then $$q \circ L : X \to \R$$ is a seminorm on $$X.$$ The seminorm $$q \circ L$$ will be a norm on $$X$$ if and only if $$L$$ is injective and the restriction $$q\big\vert_{L(X)}$$ is a norm on $$L(X).$$

Minkowski functionals and seminorms
Seminorms on a vector space $$X$$ are intimately tied, via Minkowski functionals, to subsets of $$X$$ that are convex, balanced, and absorbing. Given such a subset $$D$$ of $$X,$$ the Minkowski functional of $$D$$ is a seminorm. Conversely, given a seminorm $$p$$ on $$X,$$ the sets$$\{x \in X : p(x) < 1\}$$ and $$\{x \in X : p(x) \leq 1\}$$ are convex, balanced, and absorbing and furthermore, the Minkowski functional of these two sets (as well as of any set lying "in between them") is $$p.$$

Algebraic properties
Every seminorm is a sublinear function, and thus satisfies all properties of a sublinear function, including convexity, $$p(0) = 0,$$ and for all vectors $$x, y \in X$$: the reverse triangle inequality: $$|p(x) - p(y)| \leq p(x - y)$$ and also $0 \leq \max \{p(x), p(-x)\}$ and $$p(x) - p(y) \leq p(x - y).$$

For any vector $$x \in X$$ and positive real $$r > 0:$$ $$x + \{y \in X : p(y) < r\} = \{y \in X : p(x - y) < r\}$$ and furthermore, $$\{x \in X : p(x) < r\}$$ is an absorbing disk in $$X.$$

If $$p$$ is a sublinear function on a real vector space $$X$$ then there exists a linear functional $$f$$ on $$X$$ such that $$f \leq p$$ and furthermore, for any linear functional $$g$$ on $$X,$$ $$g \leq p$$ on $$X$$ if and only if $$g^{-1}(1) \cap \{x \in X : p(x) < 1\} = \varnothing.$$

Other properties of seminorms

Every seminorm is a balanced function. A seminorm $$p$$ is a norm on $$X$$ if and only if $$\{x \in X : p(x) < 1\}$$ does not contain a non-trivial vector subspace.

If $$p : X \to [0, \infty)$$ is a seminorm on $$X$$ then $$\ker p := p^{-1}(0)$$ is a vector subspace of $$X$$ and for every $$x \in X,$$ $$p$$ is constant on the set $$x + \ker p = \{x + k : p(k) = 0\}$$ and equal to $$p(x).$$

Furthermore, for any real $$r > 0,$$ $$r \{x \in X : p(x) < 1\} = \{x \in X : p(x) < r\} = \left\{x \in X : \tfrac{1}{r} p(x) < 1 \right\}.$$

If $$D$$ is a set satisfying $$\{x \in X : p(x) < 1\} \subseteq D \subseteq \{x \in X : p(x) \leq 1\}$$ then $$D$$ is absorbing in $$X$$ and $$p = p_D$$ where $$p_D$$ denotes the Minkowski functional associated with $$D$$ (that is, the gauge of $$D$$). In particular, if $$D$$ is as above and $$q$$ is any seminorm on $$X,$$ then $$q = p$$ if and only if $$\{x \in X : q(x) < 1\} \subseteq D \subseteq \{x \in X : q(x) \leq\}.$$

If $$(X, \|\,\cdot\,\|)$$ is a normed space and $$x, y \in X$$ then $$\|x - y\| = \|x - z\| + \|z - y\|$$ for all $$z$$ in the interval $$[x, y].$$

Every norm is a convex function and consequently, finding a global maximum of a norm-based objective function is sometimes tractable.

Relationship to other norm-like concepts
Let $$p : X \to \R$$ be a non-negative function. The following are equivalent:

$$p$$ is a seminorm.

$$p$$ is a convex $F$-seminorm.

$$p$$ is a convex balanced G-seminorm.

If any of the above conditions hold, then the following are equivalent:

$$p$$ is a norm;

$$\{x \in X : p(x) < 1\}$$ does not contain a non-trivial vector subspace.

There exists a norm on $$X,$$ with respect to which, $$\{x \in X : p(x) < 1\}$$ is bounded.

If $$p$$ is a sublinear function on a real vector space $$X$$ then the following are equivalent:

$$p$$ is a linear functional;

$$p(x) + p(-x) \leq 0 \text{ for every } x \in X$$;

$$p(x) + p(-x) = 0 \text{ for every } x \in X$$;

Inequalities involving seminorms
If $$p, q : X \to [0, \infty)$$ are seminorms on $$X$$ then:

$$p \leq q$$ if and only if $$q(x) \leq 1$$ implies $$p(x) \leq 1.$$

If $$a > 0$$ and $$b > 0$$ are such that $$p(x) < a$$ implies $$q(x) \leq b,$$ then $$a q(x) \leq b p(x)$$ for all $$x \in X.$$

Suppose $$a$$ and $$b$$ are positive real numbers and $$q, p_1, \ldots, p_n$$ are seminorms on $$X$$ such that for every $$x \in X,$$ if $$\max \{p_1(x), \ldots, p_n(x)\} < a$$ then $$q(x) < b.$$ Then $$a q \leq b \left(p_1 + \cdots + p_n\right).$$

If $$X$$ is a vector space over the reals and $$f$$ is a non-zero linear functional on $$X,$$ then $$f \leq p$$ if and only if $$\varnothing = f^{-1}(1) \cap \{x \in X : p(x) < 1\}.$$

If $$p$$ is a seminorm on $$X$$ and $$f$$ is a linear functional on $$X$$ then:

$$|f| \leq p$$ on $$X$$ if and only if $$\operatorname{Re} f \leq p$$ on $$X$$ (see footnote for proof).

$$f \leq p$$ on $$X$$ if and only if $$f^{-1}(1) \cap \{x \in X : p(x) < 1 = \varnothing\}.$$

If $$a > 0$$ and $$b > 0$$ are such that $$p(x) < a$$ implies $$f(x) \neq b,$$ then $$a |f(x)| \leq b p(x)$$ for all $$x \in X.$$

Hahn–Banach theorem for seminorms
Seminorms offer a particularly clean formulation of the Hahn–Banach theorem:
 * If $$M$$ is a vector subspace of a seminormed space $$(X, p)$$ and if $$f$$ is a continuous linear functional on $$M,$$ then $$f$$ may be extended to a continuous linear functional $$F$$ on $$X$$ that has the same norm as $$f.$$

A similar extension property also holds for seminorms:

$$

 * Proof: Let $$S$$ be the convex hull of $$\{m \in M : p(m) \leq 1\} \cup \{x \in X : q(x) \leq 1\}.$$ Then $$S$$ is an absorbing disk in $$X$$ and so the Minkowski functional $$P$$ of $$S$$ is a seminorm on $$X.$$ This seminorm satisfies $$p = P$$ on $$M$$ and $$P \leq q$$ on $$X.$$ $$\blacksquare$$

Pseudometrics and the induced topology
A seminorm $$p$$ on $$X$$ induces a topology, called the, via the canonical translation-invariant pseudometric $$d_p : X \times X \to \R$$; $$d_p(x, y) := p(x - y) = p(y - x).$$ This topology is Hausdorff if and only if $$d_p$$ is a metric, which occurs if and only if $$p$$ is a norm. This topology makes $$X$$ into a locally convex pseudometrizable topological vector space that has a bounded neighborhood of the origin and a neighborhood basis at the origin consisting of the following open balls (or the closed balls) centered at the origin: $$\{x \in X : p(x) < r\} \quad \text{ or } \quad \{x \in X : p(x) \leq r\}$$ as $$r > 0$$ ranges over the positive reals. Every seminormed space $$(X, p)$$ should be assumed to be endowed with this topology unless indicated otherwise. A topological vector space whose topology is induced by some seminorm is called.

Equivalently, every vector space $$X$$ with seminorm $$p$$ induces a vector space quotient $$X / W,$$ where $$W$$ is the subspace of $$X$$ consisting of all vectors $$x \in X$$ with $$p(x) = 0.$$ Then $$X / W$$ carries a norm defined by $$p(x + W) = p(x).$$ The resulting topology, pulled back to $$X,$$ is precisely the topology induced by $$p.$$

Any seminorm-induced topology makes $$X$$ locally convex, as follows. If $$p$$ is a seminorm on $$X$$ and $$r \in \R,$$ call the set $$\{x \in X : p(x) < r\}$$ the ; likewise the closed ball of radius $$r$$ is $$\{x \in X : p(x) \leq r\}.$$ The set of all open (resp. closed) $$p$$-balls at the origin forms a neighborhood basis of convex balanced sets that are open (resp. closed) in the $$p$$-topology on $$X.$$

Stronger, weaker, and equivalent seminorms
The notions of stronger and weaker seminorms are akin to the notions of stronger and weaker norms. If $$p$$ and $$q$$ are seminorms on $$X,$$ then we say that $$q$$ is than $$p$$ and that $$p$$ is  than $$q$$ if any of the following equivalent conditions holds:

 * 1) The topology on $$X$$ induced by $$q$$ is finer than the topology induced by $$p.$$
 * 2) If $$x_{\bull} = \left(x_i\right)_{i=1}^{\infty}$$ is a sequence in $$X,$$ then $$q\left(x_{\bull}\right) := \left(q\left(x_i\right)\right)_{i=1}^{\infty} \to 0$$ in $$\R$$ implies $$p\left(x_{\bull}\right) \to 0$$ in $$\R.$$
 * 3) If $$x_{\bull} = \left(x_i\right)_{i \in I}$$ is a net in $$X,$$ then $$q\left(x_{\bull}\right) := \left(q\left(x_i\right)\right)_{i \in I} \to 0$$ in $$\R$$ implies $$p\left(x_{\bull}\right) \to 0$$ in $$\R.$$
 * 4) $$p$$ is bounded on $$\{x \in X : q(x) < 1\}.$$
 * 5) If $$\inf{} \{q(x) : p(x) = 1, x \in X\} = 0$$ then $$p(x) = 0$$ for all $$x \in X.$$
 * 6) There exists a real $$K > 0$$ such that $$p \leq K q$$ on $$X.$$

The seminorms $$p$$ and $$q$$ are called if they are both weaker (or both stronger) than each other. This happens if they satisfy any of the following conditions:

The topology on $$X$$ induced by $$q$$ is the same as the topology induced by $$p.$$

$$q$$ is stronger than $$p$$ and $$p$$ is stronger than $$q.$$

If $$x_{\bull} = \left(x_i\right)_{i=1}^{\infty}$$ is a sequence in $$X$$ then $$q\left(x_{\bull}\right) := \left(q\left(x_i\right)\right)_{i=1}^{\infty} \to 0$$ if and only if $$p\left(x_{\bull}\right) \to 0.$$

There exist positive real numbers $$r > 0$$ and $$R > 0$$ such that $$r q \leq p \leq R q.$$

Normability and seminormability
A topological vector space (TVS) is said to be a (respectively, a ) if its topology is induced by a single seminorm (resp. a single norm). A TVS is normable if and only if it is seminormable and Hausdorff or equivalently, if and only if it is seminormable and T1 (because a TVS is Hausdorff if and only if it is a T1 space). A  is a topological vector space that possesses a bounded neighborhood of the origin.

Normability of topological vector spaces is characterized by Kolmogorov's normability criterion. A TVS is seminormable if and only if it has a convex bounded neighborhood of the origin. Thus a locally convex TVS is seminormable if and only if it has a non-empty bounded open set. A TVS is normable if and only if it is a T1 space and admits a bounded convex neighborhood of the origin.

If $$X$$ is a Hausdorff locally convex TVS then the following are equivalent:

$$X$$ is normable.

$$X$$ is seminormable.

$$X$$ has a bounded neighborhood of the origin.

The strong dual $$X^{\prime}_b$$ of $$X$$ is normable.

The strong dual $$X^{\prime}_b$$ of $$X$$ is metrizable.

 Furthermore, $$X$$ is finite dimensional if and only if $$X^{\prime}_{\sigma}$$ is normable (here $$X^{\prime}_{\sigma}$$ denotes $$X^{\prime}$$ endowed with the weak-* topology).

The product of infinitely many seminormable space is again seminormable if and only if all but finitely many of these spaces trivial (that is, 0-dimensional).

Topological properties

If $$X$$ is a TVS and $$p$$ is a continuous seminorm on $$X,$$ then the closure of $$\{x \in X : p(x) < r\}$$ in $$X$$ is equal to $$\{x \in X : p(x) \leq r\}.$$

The closure of $$\{0\}$$ in a locally convex space $$X$$ whose topology is defined by a family of continuous seminorms $$\mathcal{P}$$ is equal to $$\bigcap_{p \in \mathcal{P}} p^{-1}(0).$$

A subset $$S$$ in a seminormed space $$(X, p)$$ is bounded if and only if $$p(S)$$ is bounded.

If $$(X, p)$$ is a seminormed space then the locally convex topology that $$p$$ induces on $$X$$ makes $$X$$ into a pseudometrizable TVS with a canonical pseudometric given by $$d(x, y) := p(x - y)$$ for all $$x, y \in X.$$

The product of infinitely many seminormable spaces is again seminormable if and only if all but finitely many of these spaces are trivial (that is, 0-dimensional).

Continuity of seminorms
If $$p$$ is a seminorm on a topological vector space $$X,$$ then the following are equivalent:

$$p$$ is continuous.

$$p$$ is continuous at 0;

$$\{x \in X : p(x) < 1\}$$ is open in $$X$$;

$$\{x \in X : p(x) \leq 1\}$$ is closed neighborhood of 0 in $$X$$;

$$p$$ is uniformly continuous on $$X$$;

There exists a continuous seminorm $$q$$ on $$X$$ such that $$p \leq q.$$

In particular, if $$(X, p)$$ is a seminormed space then a seminorm $$q$$ on $$X$$ is continuous if and only if $$q$$ is dominated by a positive scalar multiple of $$p.$$

If $$X$$ is a real TVS, $$f$$ is a linear functional on $$X,$$ and $$p$$ is a continuous seminorm (or more generally, a sublinear function) on $$X,$$ then $$f \leq p$$ on $$X$$ implies that $$f$$ is continuous.

Continuity of linear maps
If $$F : (X, p) \to (Y, q)$$ is a map between seminormed spaces then let $$\|F\|_{p,q} := \sup \{q(F(x)) : p(x) \leq 1, x \in X\}.$$

If $$F : (X, p) \to (Y, q)$$ is a linear map between seminormed spaces then the following are equivalent:

$$F$$ is continuous;

$$\|F\|_{p,q} < \infty$$;

There exists a real $$K \geq 0$$ such that $$p \leq K q$$;
 If $$F$$ is continuous then $$q(F(x)) \leq \|F\|_{p,q} p(x)$$ for all $$x \in X.$$
 * In this case, $$\|F\|_{p,q} \leq K.$$

The space of all continuous linear maps $$F : (X, p) \to (Y, q)$$ between seminormed spaces is itself a seminormed space under the seminorm $$\|F\|_{p,q}.$$ This seminorm is a norm if $$q$$ is a norm.

Generalizations
The concept of in composition algebras does  share the usual properties of a norm.

A composition algebra $$(A, *, N)$$ consists of an algebra over a field $$A,$$ an involution $$\,*,$$ and a quadratic form $$N,$$ which is called the "norm". In several cases $$N$$ is an isotropic quadratic form so that $$A$$ has at least one null vector, contrary to the separation of points required for the usual norm discussed in this article.

An or a  is a seminorm $$p : X \to \R$$ that also satisfies $$p(x + y) \leq \max \{p(x), p(y)\} \text{ for all } x, y \in X.$$

Weakening subadditivity: Quasi-seminorms

A map $$p : X \to \R$$ is called a if it is (absolutely) homogeneous and there exists some $$b \leq 1$$ such that $$p(x + y) \leq b p(p(x) + p(y)) \text{ for all } x, y \in X.$$ The smallest value of $$b$$ for which this holds is called the

A quasi-seminorm that separates points is called a on $$X.$$

Weakening homogeneity - $$k$$-seminorms

A map $$p : X \to \R$$ is called a if it is subadditive and there exists a $$k$$ such that $$0 < k \leq 1$$ and for all $$x \in X$$ and scalars $$s,$$$$p(s x) = |s|^k p(x)$$ A $$k$$-seminorm that separates points is called a  on $$X.$$

We have the following relationship between quasi-seminorms and $$k$$-seminorms:

title: Seminorm


text:
Dumnonia

Dumnonia is the Latinised name for a Brythonic kingdom that existed in Sub-Roman Britain between the late 4th and late 8th centuries CE in the more westerly parts of present-day South West England. It was centred in the area of modern Devon, but also included modern Cornwall and part of Somerset, with its eastern boundary changing over time as the gradual westward expansion of the neighbouring Anglo-Saxon kingdom of Wessex encroached on its territory. The spelling Damnonia is sometimes encountered, but that spelling is also used for the land of the Damnonii, later part of the Kingdom of Strathclyde, in present-day southern Scotland. The form Domnonia also occurs. The name of the kingdom shares a linguistic relationship with the Breton region of Domnonée (Domnonea).

Name
The kingdom is named after the Dumnonii, a British Celtic tribe living in the south-west at the time of the Roman invasion of Britain, according to Ptolemy's Geography. Variants of the name Dumnonia include Domnonia and Damnonia, the latter being used by Gildas in the 6th century as a pun on "damnation" to deprecate the area's contemporary ruler Constantine. The name etymologically originates from Proto-Celtic *dubno- '(adjective) deep; (noun) world'. Groups with similar names existed in Scotland (Damnonii) and Ireland (Fir Domnann). Later, the area became known to the English of neighbouring Wessex as the kingdom of West Wales, and its inhabitants were also known to them as Defnas (i.e. men of Dumnonia). In Welsh, and similarly in the Southwestern Brythonic languages, it was Dyfneint and this is the form which survives today in the name of the county of Devon (Modern Dyfnaint, Dewnans, Devnent).

There is evidence, based on an entry in the Ravenna Cosmography, that there may have been a sub-tribe in the western part of the territory known as the Cornovii from whose name the first element of the present-day name of Cornwall is probably derived.

Following a period of emigration from south-western Britain to north-western Gaul (Armorica) in the 5th and 6th centuries, a sister kingdom (Domnonée in modern French), was established on the north-facing Atlantic coast of the continent in the region that was to become known as Brittany. Historian Barbara Yorke has speculated that the Dumnonii may have seen the end of the Roman empire as an opportunity to establish control in new areas. Before the arrival of the Romans, the Dumnonii seem to have inhabited the south-west peninsula of Britain as far east as the River Parrett in Somerset and the River Axe in Dorset, judging by the coin distributions of the Dobunni and Durotriges. In the Roman period there was a provincial boundary between the area governed from Exeter and those governed from Dorchester and Ilchester. Julius Caesar's Commentarii de Bello Gallico, Book III notes the close trading and military relationship between the continental Veneti of Armorica and the south-western insular British.

In the post Roman period the kingdom of Dumnonia covered Cornwall, Devon and parts of west Somerset. It had close cultural and religious links with Brittany, Wales and Ireland.

Culture and industries
The cultural connections of the pre-Roman Dumnonii, as expressed in their ceramics, are thought to have been with the peninsula of Armorica across the Channel, and with Wales and Ireland, rather than with the south-east of Britain. The people of Dumnonia would have spoken a Brythonic dialect, the ancestor of modern Cornish and Breton. Irish immigrants, the Déisi, are evidenced by the inscribed stones they have left behind—sometimes written in Ogham, sometimes in Latin, sometimes in both, confirmed and supplemented by place-name studies.

Apart from fishing and agriculture, the main economic resource of the Dumnonii was tin mining, the tin having been exported since ancient times from the port of Ictis (St Michael's Mount or Mount Batten). Tin working continued throughout Roman occupation and appears to have reached a peak during the 3rd century CE. The area maintained trade links with Gaul and the Mediterranean after the Roman withdrawal, and it is likely that tin played an important part in this trade. Post-Roman imported pottery has been excavated from many sites across the region. An apparent surge in late-5th-century Mediterranean imports is thought to be related to the trade in metals from Cornwall and Wales to the Byzantine Empire.

Christianity seems to have survived in Dumnonia after the Roman departure from Britain, with a number of late Roman Christian cemeteries extending into the post-Roman period. In the 5th and 6th centuries the area was allegedly evangelised by the children of Brychan and saints from Ireland, like Saint Piran; and Wales, like Saint Petroc or Saint Keyne. There were important monasteries at Bodmin and Glastonbury; and also Exeter where 5th-century burials discovered near the cathedral probably represent the cemetery of the foundation attended by Saint Boniface (although whether this was Saxon or Brythonic is somewhat controversial). Sporadically, Cornish bishops are named in various records until they submitted to the See of Canterbury in the mid-9th century. Parish organisation was a later development of fully Normanised times.

Settlements
Around 55 CE, the Romans established a legionary fortress at Isca Dumnoniorum, modern Exeter, but west of Exeter the area remained largely un-Romanised. Most of Dumnonia is notable for its lack of a villa system – though there were substantial numbers south of Bath and around Ilchester –, and for its many settlements that have survived from the Romano-British period. As in other Brythonic areas, Iron Age hillforts, such as Hembury and Cadbury Castle, were refortified in post-Roman times for the use of chieftains or kings, and other high-status settlements such as Tintagel seem to have been reconstructed during the period. Local archaeology has revealed that the isolated enclosed farmsteads known locally as rounds seem to have survived the Roman departure from Britain; but they were subsequently replaced, in the 6th and 7th centuries, by unenclosed farms taking the Brythonic toponymic tre(f)-.

Exeter, called Caer Uisc in Brythonic, was later the site of an important Saxon minster, but was still partially inhabited by Dumnonian Britons until the 10th century when Æthelstan expelled them. By the mid-9th century, the royal seat may have been relocated further west, during the West Saxon advance, to Lis-Cerruyt (modern Liskeard). Cornish earls in the 10th century were said to have moved to Lostwithiel after Liskeard was seized. It has been suggested that the rulers of Dumnonia were itinerant, stopping at various royal residences, such as Tintagel and Cadbury Castle, at different times of the year, and possibly simultaneously holding lands in Brittany across the Channel. There is textual and archaeological evidence that districts such as Trigg were used as marshalling points for "war hosts" from across the region.

History and rulers
Although subjugated by about 78 CE, the local population could have retained strong local control, and Dumnonia may have been self-governed under Roman rule. Geoffrey of Monmouth stated that the ruler of Dumnonia, perhaps about the period c. 290 – c.  305, was Caradocus. If not an entirely legendary figure, Caradocus would not have been a king in the true sense but may have held a powerful office within the Roman administration.

The post-Roman history of Dumnonia comes from a variety of sources and is considered exceedingly difficult to interpret given that historical fact, legend and confused pseudo-history are compounded by a variety of sources in Middle Welsh and Latin. The main sources available for discussion of this period include Gildas's De Excidio Britanniae and Nennius's Historia Brittonum, the Annales Cambriae, Anglo-Saxon Chronicle, William of Malmesbury's Gesta Regum Anglorum and De Antiquitate Glastoniensis Ecclesiae, along with texts from the Black Book of Carmarthen and the Red Book of Hergest, and Bede's Historia ecclesiastica gentis Anglorum as well as "The Descent of the Men of the North" (Bonedd Gwŷr y Gogledd, in Peniarth MS 45 and elsewhere) and the Book of Baglan.

Conflict with the Saxons
In 577 Ceawlin of Wessex's victory at the Battle of Deorham caused the Britons of Dumnonia to be cut off by land from their Welsh allies, but since sea travel was not difficult this may not have been a severe loss. Clemen is thought to have been king when the Britons fought the Battle of Beandun in 614. This is most likely to have been at Bindon near Axmouth in Devon. Bampton, Oxfordshire has also been proposed as the site, but the claim lacks evidence.

According to the Flores Historiarum, attributed incorrectly to Matthew of Westminster, the Britons were still in possession of Exeter in 632, when it was bravely defended against Penda of Mercia until relieved by Cadwallon, who engaged and, according to Geoffrey of Monmouth, defeated the Mercians with "great slaughter to their troops". However, since the late 19th century this siege has not been considered a historical fact.

Around 652 Cenwalh of Wessex made a breakthrough against the Dumnonian defensive lines at the battle of Bradford-upon-Avon. The West Saxon victory at the Battle of Peonnum (possibly modern Penselwood in east Somerset), around 658, resulted in the Saxons capturing "as far as the Parrett" and the eastern part of Dumnonia being permanently annexed by Wessex.

The entry for 661 in Æthelweard's translation of the Anglo-Saxon Chronicle into Latin, known as the Chronicon Æthelweardi, describes Cenwalh of Wessex fighting a battle at Posentesburh. Though it appears from the context that this is a battle against Wulfhere of Mercia (which he may have lost), if Posentesburh is identified with Posbury, near Crediton, Devon, then some conflict with the Britons can be postulated. In Willibald's Life of Saint Boniface the head of Examchester monastery, which can be identified with Exeter, Devon, has a Germanic name (Wulfhard) during the time Boniface studied there. Boniface self-identifies as Anglo-Saxon by birth (using Anglorum in his letter to the English people) and therefore Exeter may have been under West Saxon control at this time, that is, the late 7th century. At this time Dumnonia was sufficiently part of the known world for Aldhelm, later bishop of Sherborne, to address a letter around 705, to its king Geraint regarding the date of Easter. In 682 Wessex forces "advanced as far as the sea", but it is unclear where this was. In 705 a bishopric was set up in Sherborne for the Saxon area west of Selwood.

In 710 Geraint was defeated in battle by King Ine of Wessex, but in 722 the Annales Cambriae claim a victory by the British in Cornwall at Hehil. By about 755, the territory of the "Defnas" was coming under significant pressure from the Saxon army. The campaigns of Egbert of Wessex in Devon between 813 and 822 probably signalled the conquest of insular Dumnonia leaving a rump state in what is today called Cornwall, known at the time as Cerniu, Cernyw, or Kernow, and to the Anglo-Saxons as Cornwall or "West Wales".

In 825 a battle was fought between the "Welsh", presumably those of Dumnonia, and the Anglo-Saxons. The Anglo-Saxon Chronicle states: "We fought the Wealas (Cornish) and the Defnas (Devonians) at Gafulforda" (perhaps Galford in west Devon). However, there is no mention of who won or who lost. A further rebellion in 838, when the "West Welsh" were supported by Danish forces, was crushed by Egbert at the battle of Hingston Down.

The Cornish bishop of Bodmin acknowledged the authority of Canterbury in 870 and the last-known Cornish king, Donyarth, died in 875. By the 880s Wessex had gained control of at least part of Cornwall, where Alfred the Great had estates. In about 936, according to William of Malmesbury writing around 1120, Athelstan evicted the Britons from Exeter and the rest of Devon, and set the east bank of the River Tamar as Cornwall's border.

Although the chronology of Wessex expansion into all of Dumnonia is unclear, Devon had long been absorbed into England by the reign of Edward the Confessor. The early-12th-century Gesta Herewardi gives the King of Cornwall just before the Norman Conquest as a man named Alef.

Dumnonian continuity in Cornwall and Brittany
Two waves of migrations took place to Armorica (Brittany) from Dumnonia. Some histories propose the theory that this may have resulted in rulers who exercised kingship in both Brittany and Dumnonia, explaining those occurrences of the same names of rulers in both territories. There are also numerous correspondences of Insular Celtic saints, and place names and a close linguistic relationship between Cornish (Kernowek) and Breton (Brezhoneg). However, the Breton regions of Kernev/Cornouaille (Cornwall) and Domnonée (Devon) have well-established histories including entirely separate rulers from Dumnonia in Britain (see Duchy of Brittany).

While Cornwall retained its language and culture, Devon's had significantly diminished by the arrival of the Saxon invaders in the 7th century, almost entirely due to the large-scale migration of Britons from greater Dumnonia to Armorica at the end of the Roman occupation. J.B. Gover wrote in 1931 that"by the middle of the seventh century Devon was a sparsely settled Celtic kingdom due to large-scale emigration to Armorica a century and more earlier, and that once the resistance of its kings had been broken down no considerable native population remained to complicate the life of the new settlers."The relationship between the new Saxon overlords and the remaining indigenous Britons appears to have been peaceable and many Celtic place-names survive in the county, although not to the extent of that of the neighbouring sub-tribe, the Cornovii, who became modern-day Cornwall.

The pre-medieval region of Cornouaille (Breton: Kernev) in the Brittany region of the Armorican peninsula is assumed to owe its name to descendants originating in insular Cornwall. The territories of the ancient Cornouaille region coincide mostly with the southern part of the French departement of the Finistère, and some of its territorial lands are included in the departements of Côtes d'Armor and Morbihan. At least part of the original territory associated with the pre-medieval Breton kingdom of Domnonea, coincides with the modern French department of Côtes-d'Armor.

There is debate about the location of Arthur's supposed great victory at the Battle of Mount Badon, where the Britons fought off Anglo-Saxons. Most historians believe this battle, if it was historical, was fought outside the territory, at Bath, for instance. Geoffrey of Monmouth claimed that Arthur's final Battle of Camlann was fought in Cornwall: tradition points to Slaughterbridge, near Camelford, which itself has been claimed, without foundation, to be the location of Camelot.

title: Dumnonia