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In mathematics, any vector space V has a corresponding dual vector space consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, to avoid ambiguity may be called the algebraic dual space; when defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces; when applied to vector spaces of functions, dual spaces are used to describe measures and Hilbert spaces. The dual space is an important concept in functional analysis. Given any vector space V over a field F, the dual space V∗ is defined as the set of all linear maps φ: V → F. Since linear maps are vector space homomorphisms, the dual space is sometimes denoted by Hom.

The dual space V∗ itself becomes a vector space over F when equipped with an addition and scalar multiplication satisfying: = φ + ψ = a for all φ and ψ ∈ V∗, x ∈ V, a ∈ F. Elements of the algebraic dual space V ∗ are sometimes called one-forms; the pairing of a functional φ in the dual space V∗ and an element x of V is sometimes denoted by a bracket: φ = or φ = ⟨φ,x⟩. This pairing defines a nondegenerate bilinear mapping ⟨·,·⟩: V∗ × V → F called the natural pairing. If V is finite-dimensional V∗ has the same dimension as V. Given a basis in V, it is possible to construct a specific basis in V∗, called the dual basis; this dual basis is a set of linear functionals on V, defined by the relation e i = c i, i = 1, …, n for any choice of coefficients ci ∈ F. In particular, letting in turn each one of those coefficients be equal to one and the other coefficients zero, gives the system of equations e i = δ j i where δ j i is the Kronecker delta symbol; this property is referred to as biorthogonality property.

For example, if V is R2, let its basis be chosen as. Note that the basis vectors are not orthogonal to each other. E1 and e2 are one-forms such that e1 = 1, e1 = 0, e2 = 0, e2 = 1. We can express this system of equations using matrix notation as =. Solving this equation, we find the dual basis to be. Recalling that e1 and e2 are functionals, we can rewrite them as e1 = e2 = − x + y. In general, when V is Rn, if E = is a matrix whose columns are the basis vectors and Ê = is a matrix whose columns are the dual basis vectors E T E ^ = I n, where In is an identity matrix of order n; the biorthogonality property of these two basis sets allows us to represent any point x in V as x = ∑ i ⟨ x, e i ⟩ e i = ∑ i ⟨ x, e i ⟩ e i when the basis ve

Cookie Lommel is an American author, film producer, activist. Cookie Lommel was born in Ohio, she began her career in the entertainment industry in 1981. Early on she worked as a journalist for Cashbox magazine and Radio & Records magazine, she has worked as an entertainment industry reporter at CNN and as entertainment editor at Teen Magazine. In 2003, she was named the executive director of the Jewish Labor Committee's Western region. Prior to this, in 1992 Lommel founded Operation Unity, a non-profit organization that funded the travel of inner city minority students to live on an Israeli Kibbutz. Cookie Lommel authored the 2001 book The History of Rap Music, Black Filmmakers in 2002, she is the author of unauthorized biographies for Russell Simmons, Arnold Schwarzenegger, James Oglethorpe, Johnnie L. Cochran, Jr. Arthur Miller, Robert Church, Madame C. J. Walker, Michelle Pfeiffer, Mary Church Terrell, she was married to actor and film director Ulli Lommel between 1988 and his death in 2017

Tongrenelle is a hamlet in the municipality of Sombreffe, Wallonia Region, Belgium. It is east of the village of Ligny and north of the village of Boignée and on the south-eastern side of the village of Tongrinne. Within the hamlet is the Ferme-Château de Tongrenelle. A castle existed on the site which entered the historical record in 1209. During the Battle of Ligny 16 June 1815 the castle, a formidable redoubt, was garrisoned by a contingent from the 3rd Battalion of the Prussian 27th Regiment. During the first half of the 19th century the castle fell into disrepair and was demolished in 1860. What remains is now a moated manor house with buildings dating from the 17th and 18th centuries. "Ferme-Château de Tongrenelle", Sambre Orneau, 18 April 2014, retrieved 23 October 2016 Siborne, The Waterloo Campaign, 1815, Westminster: A. Constable Uffindell, The Eagle's Last Triumph: Napoleon at Ligny, June 1815, The History Press, p. ~224, ISBN 9780750967020

George W. Walker was an industrial and automotive designer, his most notable work was the original Ford Thunderbird. His father worked for the Erie Railroad and the family moved several times, settling in Cleveland, Ohio when Walker was in his teens, he played semi-professional football and held down odd jobs, but his interest in art led to art school in Los Angeles. He began his professional career as an illustrator for department store advertising as a student in LA and as an independent in Cleveland, he broke into the auto business doing illustration work for the failing Peerless automobile company in the late 1920s. He went on to a brief stint working for Harley Earl and John Tjaarda at General Motors to Graham-Paige. In 1929 the stock market crash spelled the end for many companies, including Graham-Paige, Walker went looking for work, he found it with Dura. This company supplied several automakers with parts; this job led to contact with the primary designer for Ford Motor Company. Walker's firm did substantial design work for Ford parts in the late 1940s began styling work for some Ford cars.

In the early 1950s he joined Ford at the behest of Ford executive Ernie Breech, bringing colleagues Elwood Engel and Joe Oros. Walker became corporate vice-president of Ford Motor Company for design in 1955, he stepped down from all his positions at Ford in 1961 after reaching the company's mandatory retirement age of 65. Walker appeared on the November 1957 cover of TIME Magazine. On October 15, 1959, he appeared. Walker's career included industrial design for clocks, bread boxes, chemistry sets and roller skates, among other products, he moved to Gulf Stream, where he became mayor in 1976. He died on January 19, 1993 in Tucson, Arizona at the age of 96. George Walker oral history interview TBird retrospective

Arthur Bell Nicholls was the husband of the English novelist Charlotte Brontë. Between 1845 and 1861 Nicholls was one of Patrick Brontë's curates and was married to his eldest surviving child, for the last nine months of her life, he cared for Patrick Brontë after Charlotte Brontë's death and spent the rest of his life in the shadow of her reputation. He returned to his native Ireland and left the church. Nicholls was one of ten children born to William Nicholls, a Presbyterian farmer and Margaret Bell Nicholls, a member of the Anglican Church of Ireland in Killead, County Antrim, in Ireland, he was educated at the Royal Free School in Banagher, County Offaly, whose headmaster was his uncle, Alan Bell. In 1836 Nicholls entered Trinity College, from where he graduated in 1844. Nicholls was ordained as a deacon in 1845 in Lichfield and became Patrick Brontë's curate in June that year. Charlotte Brontë said of him that he appeared to be a respectable young man who read well, she hoped that he would give satisfaction.

Although he visited the poor of the parish every afternoon, he was considered to be strict and conventional, in 1847 he carried out a campaign to prevent women from hanging their washing out to dry in the cemetery. Charlotte noted sadly that while he was away on holiday in Ireland many parishioners said that they hoped he would not return, he began to develop closer relations with Charlotte, who by that time had written Jane Eyre, they conducted a friendly exchange of letters. In December 1848, he conducted the funeral service of Emily Brontë. On 13 December 1852 Nicholls asked Charlotte for her hand in marriage. Charlotte's father vehemently refused to approve the union on the grounds that a poor Irish pastor should never be bold enough to suggest marrying his famous daughter. In 1853 Nicholls announced his intention to leave for Australia as a missionary, but he changed his mind despite collecting references and a farewell gift from the parishioners, he was re-deployed for several months to another parish, but he had several secret meetings with Charlotte in Haworth.

Little by little Charlotte became persuaded by Nicholls, in February 1854 her father gave his permission for the visits. Arthur Nicholls and Charlotte Brontë were married on 29 June 1854 in her father's church at Haworth. Patrick Brontë decided on the day of the ceremony not to attend, so Charlotte was led to the altar by Margaret Wooler, her former schoolmistress at Roe Head School, they honeymooned in Wales and Ireland before returning to live with Charlotte's father at Haworth Parsonage. Following Charlotte's sudden death, nine months in 1855, Nicholls became the copyright holder of her works, making him an defensive and reluctant curator of her memory until the early twentieth century. Public interest in his wife, which began with the pseudonymously published Jane Eyre in 1847 and the public revelation of her true identity in 1850, rocketed in the months after the announcement of her death; as press speculation about Charlotte's private life became more intense and inaccurate Patrick Brontë requested the help of Charlotte's friend the novelist Elizabeth Gaskell to correct the distortions in the form of an authorised biography.

Arthur Nicholls was reluctant to participate as it would require him granting Gaskell permission to quote directly from Charlotte's personal letters. He relented but soon after the publication of The Life of Charlotte Brontë in 1857 he became embroiled in its controversies, writing furious letters to newspapers to defend Gaskell's depiction of Charlotte Brontë's miserable school days against the teachers who now felt slandered. Further allegations of slander and libel led to the biography being withdrawn and re-issued twice with corrections which only served to stoke the public's imagination. Nicholls continued living at Haworth Parsonage as Patrick's assistant until Patrick's death in June 1861 and although he was expected to succeed him as the incumbent minister the church trustees voted against him and he resigned, he put the contents of Haworth Parsonage up for auction in October 1861, retaining the family's manuscripts and private effects and distributing keepsakes to the family's servants, moved back to Ireland.

Ellen Nussey, a friend of Charlotte's, accused Nicholls of being "that wicked man, the death of dear Charlotte". Another of Charlotte's friends, Mary Taylor, reproached Ellen Nussey for exerting pressure on Charlotte to "give up her choice in a matter so important". Elizabeth Gaskell judged him intransigent and bigoted, however, that Charlotte "would never have been happy but with an exacting, law-giving, passionate man"; the two servants at the parsonage in Haworth, Tabitha Aykroyd and Martha Brown, believed that Charlotte and Arthur were happy together. During her honeymoon Charlotte wrote to Ellen Nussey: I think those married women who indiscriminatingly urge their acquaintance to marry – much to blame. For my part – I can only say with deeper sincerity and fuller significance – what I have always said in theory – Wait God's will. Indeed -- indeed Nell -- it is a solemn and perilous thing for a woman to become a wife. Man's lot is far – far different. However, on 26 December 1854 Charlotte wrote that Arthur "is my dear boy, he is dearer to me today than he was six months ago".

After the deaths of Charlotte and Patrick Brontë, Nicholls returned to Banagher in County Offaly, to live with his widowed aunt and her daughter, Mary Anna Bell, whom he married in 1864. He left the curacy and managed a small farm, refusing to co-operate with would-be biographers who wanted to exploit his connection to the Brontës. By they were interna

Rainbow Ends is the fifth studio album by Emitt Rhodes. It was released on 26 February 2016, some 43 years after his previous album, Farewell to Paradise. Album produced by Chris Price, it features contributions from Roger Joseph Manning, Jr. Jason Falkner, Nels Cline, Aimee Mann, Jon Brion, Richard Thompson, Susanna Hoffs and members of Brian Wilson’s band; the first single, "Dog on a Chain", featuring harmonies by Mann and a solo by Brion was premiered by The Wall Street Journal's blog Speakeasy in November 2015. All songs composed by Emitt Rhodes except. Rainbow Ends has received positive reviews. On the aggregate website Metacritic, the album has received a score of 81 out of a possible 100 with five reviews, indicating "universal acclaim". Mark Deming of AllMusic says it is "a mature, introspective work from a man looking for answers to the questions of life and love, it's a brave and genuinely impressive return to the spotlight from a major talent." Hal Horowitz of American Songwriter wrote that the songs on the album are "just a notch below Rhodes’ earlier work and may yet become as well-regarded as those songs."

In a 7 out of 10 review for UNCUT, Rob Hughes writes that the album "Rainbow Ends is an intensely personal vision. Indeed, it feels more like a companion piece to his great ’70s work than it does a postscript..... if it lapses into self-pity, there’s a confessional aspect that feels unnervingly candid"Rainbow Ends debuted on the Billboard 200 at #150 for the week of March 19, 2016. It placed at number 27 on the Independent Albums chart and number 46 on the Top Internet Albums chart, it is Rhodes' first charting album since 1971's Mirror, as his following album, 1973's Farewell To Paradise, never charted