Hahn-Banach theorem

The Hahn-Banach theorem (after Hans Hahn and Stefan Banach ) from the mathematical branch of functional analysis is one of the starting points of the functional analysis. It ensures the existence of a sufficient number of continuous , linear functionals on normalized vector spaces or, more generally, on locally convex spaces . The investigation of a space with the help of the continuous, linear functionals defined on it leads to a far-reaching theory of duality , which is not possible in this form on general topological vector spaces , since a statement analogous to Hahn-Banach's theorem does not apply there.

In addition, Hahn-Banach's theorem is the basis for many non-constructive proofs of existence such as B. in the separating sentence or in the sentence of Kerin-Milman .

The theorem was essentially proven by Eduard Helly as early as 1912 . Hahn does not mention Helly in his 1927 paper, but does mention Banach in his 1929 paper, if not in connection with the sentence itself. Both use Helly's inequality. The naming after Hahn and Banach first appeared in a work by Frederic Bohnenblust and A. Sobcyzk, who transferred the sentence to complex spaces. Another proof of Hahn-Banach's theorem, which does not use Helly's inequality, was given by Jean Dieudonné in 1941 .

Finite-dimensional case

If one represents vectors of a finite-dimensional real or complex vector space with respect to a fixed base in the form of a line vector , then the respective -th entries of these line vectors can be used as functions ${\displaystyle X}$ ${\displaystyle (v_{1},\ldots ,v_{n})}$ ${\displaystyle i}$

${\displaystyle x_{i}\colon X\to \mathbb {K} ,\quad (v_{1},\ldots ,v_{n})\mapsto v_{i}}$

grasp (where the basic body or ). An essential part of the importance of such a coordinate representation known from linear algebra lies in the fact that two vectors are equal if and only if all of their coordinates match: ${\displaystyle \mathbb {K} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$

${\displaystyle v=w\iff x_{i}(v)=x_{i}(w)\ \mathrm {f{\ddot {u}}r} \ i=1,\ldots ,n.}$

The coordinate functions therefore separate the points; H. are different vectors, then there is an index such that is. They are continuous linear functionals on the coordinate space . ${\displaystyle v\neq w}$ ${\displaystyle i}$ ${\displaystyle x_{i}(v)\neq x_{i}(w)}$ ${\displaystyle x_{i}}$

In infinite-dimensional spaces there is generally no construction comparable to the coordinates if one insists on the continuity of the coordinates. Hahn-Banach's theorem implies, however, that the set of all continuous linear functionals on a normalized space (or more generally on a locally convex space ) separates the points. ${\displaystyle x_{i}}$

formulation

Let it be a vector space over . ${\displaystyle X}$ ${\displaystyle \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}}$

Let it be now

• ${\displaystyle Y\subseteq X}$a linear subspace ;
• ${\displaystyle p\colon X\to \mathbb {R} }$a sublinear map ;
• ${\displaystyle f\colon Y\to \mathbb {K} }$a linear functional that applies to all . ${\displaystyle \operatorname {Re} f(y)\leq p(y)}$ ${\displaystyle y\in Y}$

Then there is a linear functional such that ${\displaystyle F\colon X\to \mathbb {K} }$

• ${\displaystyle F|_{Y}=f\,\,}$ and
• ${\displaystyle \operatorname {Re} F(x)\leq p(x)}$

applies to all . ${\displaystyle x\in X}$

The proof of this basic theorem is not constructive. One considers the set of all continuations of on with subspaces for which holds for all . Then one shows with Zorn's lemma that the set of all such continuations has maximum elements and that such a maximum element is a sought-after continuation . ${\displaystyle g\colon Z\rightarrow {\mathbb {K} }}$ ${\displaystyle f}$ ${\displaystyle Z}$ ${\displaystyle Y\subset Z\subset X}$ ${\displaystyle \operatorname {Re} g(z)\leq p(z)}$ ${\displaystyle z\in Z}$ ${\displaystyle F\colon X\to {\mathbb {K} }}$

Corollaries

Often one of the following statements, which can easily be derived from the above sentence, is meant when the "Hahn-Banach sentence" is quoted:

• If a normalized space is, there is for each a linear functional with norm 1, for which applies. If there are different points, the above-mentioned property of point separation is obtained by applying this to . ${\displaystyle X\neq \{0\}}$ ${\displaystyle x\in X}$ ${\displaystyle f}$ ${\displaystyle f(x)=\|x\|}$ ${\displaystyle x,y\in X}$ ${\displaystyle x-y\neq 0}$
• If, more generally, a normalized space, a subspace, and is not at the end of , there is a linear functional with norm 1, which vanishes at and for which applies. ${\displaystyle X}$ ${\displaystyle U}$ ${\displaystyle x\in X}$ ${\displaystyle U}$ ${\displaystyle f}$ ${\displaystyle U}$ ${\displaystyle f(x)={\text{dist}}(x,U)}$
• If a normalized space, a subspace and a continuous linear functional is on , then the same norm can be continued completely to a continuous linear functional . In other words: the restriction of functionalities is a surjective mapping of the dual spaces . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f}$ ${\displaystyle Y}$ ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle X'\to Y'}$
• If a normalized space, then a subspace is dense in if and only if it follows from and always . ${\displaystyle X}$ ${\displaystyle U\subset X}$ ${\displaystyle X}$ ${\displaystyle x'\in X'}$ ${\displaystyle x'|_{U}=0}$ ${\displaystyle x'=0}$
• Further conclusions of a geometric nature can be found in the article Separation Theorem .

literature

• Hans Hahn: About linear systems of equations in linear spaces . In: Journal for pure and applied mathematics 157 (1927), p. 214-229.
• Stefan Banach: Sur les fonctionelles linéaires I . In: Studia Mathematica 1 (1929), p. 211-216. Available for download on IMPAN.pl
• Stefan Banach: Sur les fonctionnelles linéaires II . In: Studia Mathematica 1 (1929), p. 223-239. Available for download on IMPAN.pl
• R. Meise, D. Vogt: Introduction to Functional Analysis, Vieweg, 1992

Individual evidence

1. ↑ Helly, About linear functional operators, session reports Akad. Wiss. Vienna, Volume 121, 1912, pp. 265-297
2. ^ Harry Hochstadt : Eduard Helly, father of the Hahn-Banach theorem, The Mathematical Intelligencer, Volume 2, 1980, No. 3, pp. 123–125. According to Hochstadt, Helly's proof is completely modern in form and identical to the standard proof.
3. ↑ Helly used Hahn-Banach's theorem as a lemma for a proof of a Riesz theorem, to which Banach referred in the reference to Helly.
4. ↑ Bohnenblust, Sobcyzk, Extensions of functionals on complete linear spaces, Bull. AMS, Volume 44, 1938, pp. 91-93. They point out that their proof is identical to that of Francis J. Murray of 1936 (Murray, Linear transformations in , p> 1, Trans. AMS, Volume 39, 1936, pp. 83-100), who in turn refers to Banach but does not speak of Hahn-Banach's theorem. ${\displaystyle L_{p}}$
5. ↑ Dieudonné, Sur le Théoréme de Hahn-Banach, La Rev. Sci. 79, 1941, pp. 642-643.
6. ↑ Dirk Werner: functional analysis, Springer, 2000, corollary III.1.9