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In functional analysis, it is often convenient to define a linear transformation on a complete, normed vector space by first defining a linear transformation on a dense subset of and then continuously extending to the whole space via the theorem below. The resulting extension remains linear and bounded, and is thus continuous, which makes it a ...
Pavel Urysohn. In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem or Urysohn-Brouwer lemma [1]) states that any real-valued, continuous function on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary.
The theorem is named for the mathematicians Hans Hahn and Stefan Banach, who proved it independently in the late 1920s.The special case of the theorem for the space [,] of continuous functions on an interval was proved earlier (in 1912) by Eduard Helly, [1] and a more general extension theorem, the M. Riesz extension theorem, from which the Hahn–Banach theorem can be derived, was proved in ...
If Y is an injective Banach space, then for every Banach space X, every continuous linear operator from a vector subspace of X into Y has a continuous linear extension to all of X. [ 1 ] In 1953, Alexander Grothendieck showed that any Banach space with the extension property is either finite-dimensional or else not separable .
Say that a vector subspace of has the extension property if any continuous linear functional on can be extended to a continuous linear functional on . [13] Say that X {\displaystyle X} has the Hahn-Banach extension property ( HBEP ) if every vector subspace of X {\displaystyle X} has the extension property.
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem [1] (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.
A proof (of a stronger version) of the trace theorem for Lipschitz domains can be found in Gagliardo. [1] On a -domain, the trace operator can be defined as continuous linear extension of the operator : (¯) ()
Hahn–Banach dominated extension theorem [18] (Rudin 1991, Th. 3.2) — If : is a sublinear function, and : is a linear functional on a linear subspace which is dominated by p on M, then there exists a linear extension : of f to the whole space X that is dominated by p, i.e., there exists a linear functional F such that = for all , and | | for ...