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- mathematischer Satz (de)
- teorema matemàtic (ca)
- matematisk sætning (da)
- izrek o razširitvi omejenih linearnih funkcij (sl)
- theorem on extension of bounded linear functionals (en)
- twierdzenie analizy funkcjonalnej o funkcjonałach liniowych (pl)
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- Let a scalar-valued function on a subset of a topological vector space
Then there exists a continuous linear functional on extending if and only if there exists a continuous seminorm on such that
for all positive integers and all finite sequences of scalars and elements of (en)
- Suppose is a seminorm on a vector space over the field which is either or
If is a linear functional on a vector subspace such that
then there exists a linear functional such that (en)
- Let be a vector subspace of a locally convex topological vector space and be a non-empty open convex subset disjoint from Then there exists a continuous linear functional on such that for all and on (en)
- Every continuous linear functional defined on a vector subspace of a normed space has a continuous linear extension to all of that satisfies (en)
- Let be a sublinear function on a real vector space let be any subset of and let be map.
If there exist positive real numbers and such that
then there exists a linear functional on such that on and on (en)
- Let be a sublinear function on a real vector space let be a linear functional on a vector subspace of such that on and let be any subset of
Then there exists a linear functional on that extends satisfies on and is maximal on in the following sense: if is a linear functional on that extends and satisfies on then on implies on (en)
- Let and be convex non-empty disjoint subsets of a real topological vector space
* If is open then and are separated by a closed hyperplane. Explicitly, this means that there exists a continuous linear map and such that for all If both and are open then the right-hand side may be taken strict as well.
* If is locally convex, is compact, and closed, then and are strictly separated: there exists a continuous linear map and such that for all
If is complex then the same claims hold, but for the real part of (en)
- Let be vectors in a real or complex normed space and let be scalars also indexed by
There exists a continuous linear functional on such that for all if and only if there exists a such that for any choice of scalars where all but finitely many are the following holds: (en)
- A real Banach space is reflexive if and only if every pair of non-empty disjoint closed convex subsets, one of which is bounded, can be strictly separated by a hyperplane. (en)
- Let be a vector subspace of the topological vector space and suppose is a non-empty convex open subset of with
Then there is a closed hyperplane that contains but remains disjoint from (en)
- If is a seminorm defined on a vector subspace of and if is a seminorm on such that then there exists a seminorm on such that on and on (en)
- Every continuous linear functional defined on a vector subspace of a locally convex topological vector space has a continuous linear extension to all of If in addition is a normed space, then this extension can be chosen so that its dual norm is equal to that of (en)
- If is an absorbing disk in a real or complex vector space and if be a linear functional defined on a vector subspace of such that on then there exists a linear functional on extending such that on (en)
- If and are vector spaces over the same field and if is a linear map defined on a vector subspace of then there exists a linear map that extends (en)
- Let be a sublinear function on a real or complex vector space let be any set, and let and be any maps. The following statements are equivalent:
# there exists a real-valued linear functional on such that on and on ;
# for any finite sequence of non-negative real numbers, and any sequence of elements of (en)
- Let be a sublinear function on a real vector space let a linear functional on a proper vector subspace such that on , and let be a vector in .
There exists a linear extension of such that on (en)
- Suppose is a Hausdorff locally convex TVS over the field and is a vector subspace of that is TVS–isomorphic to for some set
Then is a closed and complemented vector subspace of (en)
- Let and be non-empty convex subsets of a real locally convex topological vector space
If and then there exists a continuous linear functional on such that and for all . (en)
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- Theorem (en)
- Proposition (en)
- Lemma (en)
- Corollary (en)
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| dbp:note
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- Andenaes, 1970 (en)
- Separation of a subspace and an open convex set (en)
- The functional problem (en)
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| dbp:proof
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- Since is a complete TVS so is and since any complete subset of a Hausdorff TVS is closed, is a closed subset of
Let be a TVS isomorphism, so that each is a continuous surjective linear functional.
By the Hahn–Banach theorem, we may extend each to a continuous linear functional on
Let so is a continuous linear surjection such that its restriction to is
Let which is a continuous linear map whose restriction to is where denotes the identity map on
This shows that is a continuous linear projection onto .
Thus is complemented in and in the category of TVSs. (en)
- Given any real number the map defined by is always a linear extension of to but it might not satisfy
It will be shown that can always be chosen so as to guarantee that which will complete the proof.
If then
which implies
So define
where are real numbers.
To guarantee it suffices that because then satisfies "the decisive inequality"
To see that follows, assume and substitute in for both and to obtain
If then the right hand side equals so that multiplying by gives (en)
- The set of all possible dominated linear extensions of are partially ordered by extension of each other, so there is a maximal extension By the codimension-1 result, if is not defined on all of then it can be further extended. Thus must be defined everywhere, as claimed. (en)
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| dbp:title
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- Proof (en)
- Hahn–Banach theorem (en)
- Proof of dominated extension theorem using Zorn's lemma (en)
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| gold:hypernym
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| rdf:type
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| rdfs:label
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- Hahn–Banach theorem (en)
- Teorema de Hahn-Banach (ca)
- Hahnova–Banachova věta (cs)
- مبرهنة هان-باناخ (ar)
- Satz von Hahn-Banach (de)
- Teorema de Hahn–Banach (es)
- Théorème de Hahn-Banach (fr)
- Teorema di Hahn-Banach (it)
- ハーン–バナッハの定理 (ja)
- 한-바나흐 정리 (ko)
- Teorema de Hahn-Banach (pt)
- Stelling van Hahn-Banach (nl)
- Twierdzenie Hahna-Banacha (pl)
- Теорема Гана — Банаха (uk)
- Hahn-Banachs sats (sv)
- Теорема Хана — Банаха (ru)
- 哈恩－巴拿赫定理 (zh)
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