Statement:
Hahn-Banach Theorem: If \(V\) is a real vector space and \(p: V \rightarrow {\Bbb R}\) satisfies \(p(x+y) \le p(x) + p(y)\) and \((\forall t > 0)( p(tx) = tp(x) )\) and \(S\) is a subspace of \(V\) and \(f:S \rightarrow {\Bbb R}\) is linear and satisfies \((\forall x \in S)( f(x) \le p(x) )\) then \(f\) can be extended to \(f^{*} : V \rightarrow {\Bbb R}\) such that \(f^{*}\) is linear and \((\forall x \in V)(f^{*}(x) \le p(x))\).
Howard_Rubin_Number: 52
Parameter(s): This form does not depend on parameters
This form's transferability is: Transferable
This form's negation transferability is: Negation Transferable
Article Citations:
L o's-Ryll-Nardzewski-1951: On the application of Tychonoff's theorem in mathematical proofs
Book references
Note connections:
| Howard-Rubin Number | Statement | References |
|---|---|---|
| 52 A | On every Boolean algebra there is an additive realvalued measure. Luxemburg [1969] and Note 147. |
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| 52 B | For every proper filter \(\cal F\) of elements of aBoolean algebra \(\cal B\), there exists an additive, real valued measure\(m\) on \(\cal B\) such that \(m(x) = 1\) for all \(x\in\cal F\).Luxemburg [1969] and Note 147. |
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| 52 C | For every non-empty set X and every proper ideal\(\cal I\) of in \(\cal P(X)\) there is a measure \(m\) on \(\cal P(X)\) suchthat \(m(x) = 0\) for all \(x\in\cal I\). Luxemburg [1969] andNote 147. |
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| 52 D | Let \(\cal B_0\) be a subalgebra of a Boolean algebra\(\cal B\) and \(m_0\) an additive, real valued measure on \(\cal B_0\).Then there is a real valued, additive measure \(m\) on \(\cal B\) such that\(m = m_0\) on \(\cal B_0\) and the range of \(m\) is contained in the closedconvex hull of the range of \(m_0\). Luxemburg [1969] and Note 147. |
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| 52 E | The unit ball is convex-compact in theweak* topology in the dual of a Banach space. (See [14 Q].)Luxemburg [1969], H.~Rubin /J.~Rubin [1985] p 178,and Note 23. |
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| 52 F | If \(E\) is a topological vector space then for everycontinuous sublinear functional \(p\) on \(E\) there is a linearfunctional \(f\) on \(E\) such that \(f\le p\). Fossy/Morillon [1998]. |
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| 52 G | If \(E\) is a normed vector space then for everycontinuous sublinear functional \(p\) on \(E\) there is a linearfunctional \(f\) on \(E\) such that \(f\le p\). Fossy/Morillon [1998]. |
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| 52 H | Existence of Real Valued Banach Limits: For any directedset \((\Delta,\sqsubseteq)\) there exists a (real valued) Banach limit.Schechter [1996b] and Note 31. |
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| 52 I | Real Valued Convex Domination Theorem: For every realvector space \(X\), every subspace \(X_0\subseteq X\), every linear\(\lambda_0 :X_0\to {\Bbb R}\) and every convex \(p: X\to {\Bbb R}\), if\(\lambda_0 \le p\) on \(X_0\) then \(\lambda_0\) can be extended to a linearmap \(\lambda: X\to {\Bbb R} \) such that \(\lambda \le p\) on \(X\).Schechter [1996b] and Note 31. |
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| 52 J | Support Theorem for Real Valued Convex Functions:For every real vector space \(X\),every convex function \(p: X\to {\Bbb R}\) is a pointwise maximum ofaffine functions. Schechter [1996b] and Note 31. |
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| 52 K | Sandwich Theorem for Real Valued Functions:For every real vector space \(X\), if\(e: X\to {\Bbb R}\) is a concave function, \(g: X\to {\Bbb R}\) is a convexfunction, and \(e\le g\) everywhere on \(X\), then there exists an affinefunction \(f: X \to {\Bbb R}\) satisfying \(e \le f \le g\) everywhere on\(X\). Schechter [1996b] and Note 31. |
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| 52 L | If \(E\) is a topological vector space then for everysublinear functional \(p\) on \(E\) there is a linear functional \(f\) on \(E\)such that \(f\le p\). Fossy/Morillon [1998]. |
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| 52 M | If \(E\) is a topological vector space, \(p\)is a continuous sublinear functional on \(E\), and \(S\) is a subspace of\(E\) such that \(f\) is a linear functional on \(S\) with \(f\le p\), then\(f\) can be extended to \(f^{*} : E \rightarrow {\Bbb R}\) such that\(f^{*}\) is linear and \(f^*\le p\). Fossy/Morillon [1998]. |
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| 52 N | If \(E\) is a topological vector space, \(C\) is aan affine subspace of \(E\), and \(O\) is a non-empty open convexsubset of \(E\) such that \(C\cap O=\emptyset\), then there exists alinear functional \(f\) on \(E\) such that for all \(x\in O\), \(f(x) <\)inf\(_{z\in C}f(z)\). Dodu/Morillon [1999]. |
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| 52 O | If \(E\) is a topological vector space, \(C\) is anon-empty convex subset of \(E\), and \(O\) is a non-empty open convexsubset of \(E\) such that \(C\cap O=\emptyset\), then there exists alinear functional \(f\) on \(E\) such that for all \(x\in O\), \(f(x) <\)inf\(_{z\in C}f(z)\). Dodu/Morillon [1999]. |
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| 52 P | If \(E\) is a topological vector space, \(a\in E\),and \(O\) is a non-empty open convex subset of \(E\) such that \(a\notin O\),then there exists a linear functional \(f\) on \(E\) such that for all\(x\in O\), \(f(x) < f(a)\). Dodu/Morillon [1999]. |
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| 52 Q | If \(E\) is a topological vector space, \(C\) andand \(O\) two non-empty disjoint convex subsets of \(E\) and \(O\) is open,then there exists a linear functional \(f\) on \(E\) such that \(f[O] |
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| 52 R | If \(E\) is a topological vector space, \(C\) andand \(O\) two non-empty disjoint open convex subsets of \(E\), then thereexists a linear functional \(f\) on \(E\) such that \(f[O] < f[C]\).Dodu/Morillon [1999]. |
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| 52 S | If \(E\) is a topological vector space, \(C\) is anon-empty closed convex subset of E, and \(K\) is a non-empty compactconvex subset of \(E\), then there exists a linear functional \(f\) on \(E\)such that sup\(_{x\in K}f(x) <\) inf\(_{z\in C}f(z)\). Dodu/Morillon [1999]. |
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| 52 T | On every non-trivial Banach space thereis a non-trivial bounded linear functional. Luxemburg/V\"ath [2001] |
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