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Vector-valued Hahn–Banach theorems

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In mathematics, specifically in functional analysis and Hilbert space theory, vector-valued Hahn–Banach theorems are generalizations of the Hahn–Banach theorems from linear functionals (which are always valued in the real numbers or the complex numbers ) to linear operators valued in topological vector spaces (TVSs).

Definitions

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Throughout X and Y will be topological vector spaces (TVSs) over the field and L(X; Y) will denote the vector space of all continuous linear maps from X to Y, where if X and Y are normed spaces then we endow L(X; Y) with its canonical operator norm.

Extensions

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If M is a vector subspace of a TVS X then Y has the extension property from M to X if every continuous linear map f : MY has a continuous linear extension to all of X. If X and Y are normed spaces, then we say that Y has the metric extension property from M to X if this continuous linear extension can be chosen to have norm equal to f.

A TVS Y has the extension property from all subspaces of X (to X) if for every vector subspace M of X, Y has the extension property from M to X. If X and Y are normed spaces then Y has the metric extension property from all subspace of X (to X) if for every vector subspace M of X, Y has the metric extension property from M to X.

A TVS Y has the extension property[1] if for every locally convex space X and every vector subspace M of X, Y has the extension property from M to X.

A Banach space Y has the metric extension property[1] if for every Banach space X and every vector subspace M of X, Y has the metric extension property from M to X.

1-extensions

If M is a vector subspace of normed space X over the field then a normed space Y has the immediate 1-extension property from M to X if for every xM, every continuous linear map f : MY has a continuous linear extension such that f‖ = ‖F. We say that Y has the immediate 1-extension property if Y has the immediate 1-extension property from M to X for every Banach space X and every vector subspace M of X.

Injective spaces

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A locally convex topological vector space Y is injective[1] if for every locally convex space Z containing Y as a topological vector subspace, there exists a continuous projection from Z onto Y.

A Banach space Y is 1-injective[1] or a P1-space if for every Banach space Z containing Y as a normed vector subspace (i.e. the norm of Y is identical to the usual restriction to Y of Z's norm), there exists a continuous projection from Z onto Y having norm 1.

Properties

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In order for a TVS Y to have the extension property, it must be complete (since it must be possible to extend the identity map from Y to the completion Z of Y; that is, to the map ZY).[1]

Existence

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If f : MY is a continuous linear map from a vector subspace M of X into a complete Hausdorff space Y then there always exists a unique continuous linear extension of f from M to the closure of M in X.[1][2] Consequently, it suffices to only consider maps from closed vector subspaces into complete Hausdorff spaces.[1]

Results

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Any locally convex space having the extension property is injective.[1] 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.[1]

Theorem[1] — Suppose that Y is a Banach space over the field Then the following are equivalent:

  1. Y is 1-injective;
  2. Y has the metric extension property;
  3. Y has the immediate 1-extension property;
  4. Y has the center-radius property;
  5. Y has the weak intersection property;
  6. Y is 1-complemented in any Banach space into which it is norm embedded;
  7. Whenever Y in norm-embedded into a Banach space then identity map can be extended to a continuous linear map of norm to ;
  8. Y is linearly isometric to for some compact, Hausdorff space, extremally disconnected space T. (This space T is unique up to homeomorphism).

where if in addition, Y is a vector space over the real numbers then we may add to this list:

  1. Y has the binary intersection property;
  2. Y is linearly isometric to a complete Archimedean ordered vector lattice with order unit and endowed with the order unit norm.

Theorem[1] — Suppose that Y is a real Banach space with the metric extension property. Then the following are equivalent:

  1. Y is reflexive;
  2. Y is separable;
  3. Y is finite-dimensional;
  4. Y is linearly isometric to for some discrete finite space

Examples

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Products of the underlying field

Suppose that is a vector space over , where is either or and let be any set. Let which is the product of taken times, or equivalently, the set of all -valued functions on T. Give its usual product topology, which makes it into a Hausdorff locally convex TVS. Then has the extension property.[1]

For any set the Lp space has both the extension property and the metric extension property.

See also

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Citations

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  1. ^ a b c d e f g h i j k l m Narici & Beckenstein 2011, pp. 341–370.
  2. ^ Rudin 1991, p. 40 Stated for linear maps into F-spaces only; outlines proof.

References

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  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
  • Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
  • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.