As a consequence of the Hahn – Banach theorem, this map is injective, and isometric.

In mathematics, the Hahn – Banach theorem is a central tool in functional analysis.

" Another version of Hahn – Banach theorem is known as Hahn – Banach separation theorem or the separating hyperplane theorem, and has numerous uses in convex geometry.

It is named for Hans Hahn and Stefan Banach who proved this theorem independently in the late 1920s, although a special case was proved earlier ( in 1912 ) by Eduard Helly, and a general extension theorem from which the Hahn – Banach theorem can be derived was proved in 1923 by Marcel Riesz.

The Hahn – Banach theorem states that if is a sublinear function, and is a linear functional on a linear subspace U ⊆ V which is dominated by on U,

Another version of Hahn – Banach theorem states that if V is a vector space over the scalar field K ( either the real numbers R or the complex numbers C ), if is a seminorm, and is a K-linear functional on a K-linear subspace U of V which is dominated by on U in absolute value,

Another version of Hahn – Banach theorem is known as the Hahn – Banach separation theorem.

A famous example of a theorem of this sort is the Hahn – Banach theorem.

The Hahn – Banach theorem, is one of fundamental theorems of functional analysis.

It occurs in the proofs of several theorems of crucial importance, for instance the Hahn – Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that every nonzero ring has a maximal ideal and that every field has an algebraic closure.

This theory is due to Hans Hahn, who also showed that one obtains subfields when the number of ( non-zero ) terms is bounded by some fixed infinite cardinality.

* Huxie-Previously known as Need More, it is located somewhere between Hahn and Marble Hill.

He is also the author of the novels Die Festung ( 1995 ) ( The Fortress ), based on travels home across France in 1944, and Der Abschied ( 2000 ) ( The Parting ), about the nuclear-powered cargo vessel NS Otto Hahn.

Federally, Del Aire is located in California's 35th and 36th congressional districts, which have Cook PVIs of D + 33 and D + 11 respectively and is represented by Democrats Maxine Waters and Janice Hahn respectively.

Federally, Lennox is located in California's 36th congressional district, which has a Cook PVI of D + 11 and is represented by Democrat Janice Hahn.

Laugh-In writers included: George Schlatter, Jack Mendelsohn, Lorne Michaels, Phil Hahn, Jim Mulligan, Jack Hanrahan, Gene Farmer, Jim Abell, Bill Richmond, Don Reo, Allan Katz, Jack Wohl, Larry Siegel, John Rappaport, Allan Manings, Jack Margolis, Bob Howard, John Jay Carsey, Richard Goren ( also credited as Rowby Greeber and Rowby Goren ), Chris Bearde ( credited as Chris Beard ), Chet Dowling, David Panich, Marc London, Paul Keyes, Dave Cox, Jack Kaplan, Stephen Spears, Hugh Wedlock Jr., Coslough Johnson ( Arte Johnson's twin brother ), Hart Pomerantz, Barry Took, Digby Wolfe, Jeremy Lloyd.

Her second album, the 1978 self-titled Kate Taylor, was produced by her brother James and Lew Hahn on Columbia Records.

He belonged to an influential Los Angeles political dynasty: One brother, Gordon Hahn, replaced him on the Los Angeles City Council, while another, John, was assistant county clerk.

Armstrong as Doctor Baugh ; Fred Stewart as Reverend Tooker ; Janice Dunn as Trixie ; Seth Edwards as Sonny ; Maxwell Glanville as Lacey ; Pauline Hahn as Dixie ; Darryl Richard as Buster ; Eva Vaughn Smith as Daisy ; and Musa Williams as Sookey.

Federally, Lomita is located in California's 36th congressional district, which has a Cook PVI of D + 11 and is represented by Democrat Janice Hahn.

Federally, Marina del Rey is located in California's 36th congressional district, which has a Cook PVI of D + 12 and is represented by Democrat Janice Hahn.

* Janice Hahn, current City Councilwoman for the 15th district.

Such poets as Marilyn Chin and Li-Young Lee, Kimiko Hahn and Janice Mirikitani have also achieved prominence, as has playwright David Henry Hwang.

His sister, Janice Hahn, was reelected to a third term in 2009 as an LA City Councilwoman for the 15th District.

* Janice Hahn ( CA-36, San Pedro )

* Janice Hahn, politician, member of the Los Angeles City Council

He defeated Democrat Janice Hahn in the 1998 election with 49 % of the vote.

His daughter, Janice Hahn, sat on the Los Angeles City Council, representing San Pedro and Wilmington, and is now a member of the U. S. House of Representatives.

* Janice Hahn ( CA-36 )

In December 1938, the German chemists Otto Hahn and Fritz Strassmann sent a manuscript to Naturwissenschaften reporting they had detected the element barium after bombarding uranium with neutrons ; Lise Meitner and her nephew Otto Robert Frisch correctly interpreted these results as being nuclear fission.

A nephew, Dale Hahn, was a Superior Court judge ( in San Mateo County ) until retiring in 2004.

* James Hahn, former Mayor of Los Angeles, is a current resident.

As Mayor, Hahn appointed Bill Bratton, the former NYPD Commissioner, as Police Chief of Los Angeles and chose not to renew Bernard Parks's second term as Chief.

In the March 8, 2005 general election, Hahn placed second to Antonio Villaraigosa ; placing ahead of former Speaker of the Assembly Bob Hertzberg, Councilman Bernard Parks, State Senator Richard Alarcon, and about seven lesser known challengers.

On January 7, 2006, Hahn attended and spoke at a memorial service at Los Angeles City Hall and at a private residence for former County Federation of Labor leader Bill Robertson.

Kaku was also interviewed for two PBS documentaries produced and directed by Rosemarie Reed, a former colleague of his at WBAI Radio, The Path to Nuclear Fission: The Story of Lise Meitner and Otto Hahn, and Out from the Shadows: The Story of Irene Joliot-Curie and Frederic Joliot-Curie.

He finished with 6 percent of the primary vote, finishing behind businessman Steve Soboroff, Councilman Joel Wachs, former California State Assembly Speaker Antonio Villaraigosa, and the eventual winner, then-City Attorney James Hahn.

Kurt Hahn had been invited to address the NATO Defence College where he saw former enemies from several nations working together towards a common goal.

In 1946, Otto Hahn assumed the position of President of KWG, and in 1948, the society was renamed the Max Planck Society ( MPG ) after its former President ( 1930 – 37 ) Max Planck, who was recently deceased.

Soboroff came in third in the nonpartisan race, and Hahn and former California State Assembly Speaker Antonio Villaraigosa advanced to the runoff.

However, Mayor Hahn actually was considering the appointment before other interests persuaded him in appointing William J. Bratton, a former police commissioner of Boston and New York City, to head the department.

In 2004, former Los Angeles Mayor James K. Hahn officially branded the neighborhood as Little Ethiopia.

The Phoenix Force team has used the assistance of John Trent, Japanese-American ninjitsu master ; and Karl Hahn, former GSG 9 operator turned BND agent.

In 1994 Hahn Air was founded as independent regional carrier operating intra German routes with its base at the former military airbase Hahn, today well known as the Frankfurt Hahn

Two major Catholic apologists whom they often call Neo-Catholic, Scott Hahn and Jimmy Akin, are converts from Protestantism, and they have described EWTN, which they often call Neo-Catholic, as having a theological committee where " he majority of the committee's members at the time, none of them priests, consisted of recently converted Protestants, two of whom were former Protestant ministers ” and also as having a staff of 40 % Protestants.

** Hahn Air Base, a former frontline NATO facility near Hahn