James' theorem

In mathematics, particularly functional analysis, James' theorem, named for Robert C. James, states that a Banach space B is reflexive if and only if every continuous linear functional on B attains its maximum on the closed unit ball in B.

A stronger version of the theorem states that a weakly closed subset C of a Banach space B is weakly compact if and only if each continuous linear functional on B attains a maximum on C.

The hypothesis of completeness in the theorem cannot be dropped (James 1971).

See also

References

Khan Academy

Khan Academy is a non-profit educational organization created in 2006 by educator Salman Khan with the aim of providing a free, world-class education for anyone, anywhere. The organization produces short lectures in the form of YouTube videos. In addition to micro lectures, the organization's website features practice exercises and tools for educators. All resources are available for free to anyone around the world. The main language of the website is English, but the content is also available in other languages.

History

The founder of the organization, Salman Khan, was born in New Orleans, Louisiana, United States to immigrant parents from Bangladesh and India. After earning three degrees from the Massachusetts Institute of Technology (a BS in mathematics, a BS in electrical engineering and computer science, and an MEng in electrical engineering and computer science), he pursued an MBA from Harvard Business School.

In late 2004, Khan began tutoring his cousin Nadia who needed help with math using Yahoo!'s Doodle notepad.When other relatives and friends sought similar help, he decided that it would be more practical to distribute the tutorials on YouTube. The videos' popularity and the testimonials of appreciative students prompted Khan to quit his job in finance as a hedge fund analyst at Connective Capital Management in 2009, and focus on the tutorials (then released under the moniker "Khan Academy") full-time.

Television program

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A television program is a segment of content intended for broadcast on vision, other than a commercial, trailer, or any other segment of content not serving as attraction for viewership. It may be a single production, or more commonly, a series of related productions (also called a television series).

A television series that is intended to comprise a limited number of episodes may be called a miniseries or serial. Series without a fixed length are usually divided into seasons or series, yearly or semiannual sets of new episodes. While there is no defined length, US industry practice tends to favor longer seasons than those of some other countries.

A one-time broadcast may be called a "special," or particularly in the UK a "special episode." A television film ("made-for-TV movie" or "television movie"), is a film that is initially broadcast on television rather than released in theaters or direct-to-video.

A program can be either recorded, as on video tape, other various electronic media forms, played with an on-demand player or viewed on live television.

Fermat's Last Theorem

In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than two. The cases n = 1 and n = 2 are known to have infinitely many solutions since antiquity.

This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof it was in the Guinness Book of World Records as the "most difficult mathematical problem", one of the reasons being that it has the largest number of unsuccessful proofs.

Keke Palmer discography

The discography of Keke Palmer, an American R&B singer-songwriter, consists of one studio album, one extended play, three mixtapes and seven singles. In 2005, Palmer signed a record deal with Atlantic Records. Palmer released her debut album So Uncool on September 18, 2007. The album failed to chart on the US Billboard 200, but did chart at number 85 on the R&B chart. The album was preceded by the lead single "Keep It Movin'". In 2010, Palmer was signed by the Chairman of Interscope Records, Jimmy Iovine, and began working on an album.

In January 2011, Palmer released her first mixtape Awaken. The mixtape was officially released on January 10, 2011, for downloads on mixtape-downloading websites. The first and only single released from the mixtape was "The One You Call". A music video was also released for the song. In July 2012, Palmer released the single "You Got Me" featuring Kevin McCall. The video for the single was released on July 11, 2012. Palmer released a self-titled mixtape Keke Palmer on October 1, 2012. It includes her new singles "You Got Me" & "Dance Alone" which have already been released. On May 16, 2013, Keke Palmer released a video showing her recording and performing a cover of Alicia Keys song "If I Ain't Got You".