Vector-valued Hahn–Banach theorems

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 ℝ or the complex numbers ℂ) to linear operators valued in topological vector spaces (TVSs).

Definitions

Throughout X and Y will be topological vector spaces (TVSs) over the field 𝕂 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 norm.

Extensions
Definition: If M is a vector subspace of X then Y has the extension property from M to X if every continuous linear map f : M → Y 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||.
Definition: We say that 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.
Definition:[1] We say that Y has the extension property if for every locally convex space X and every vector subspace M of X, Y has the extension property from M to X.
Definition:[1] If Y is a Banach space then Y has the metric extension property 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
Definition: If M is a vector subspace of normed space X over the field 𝕂 then a normed space Y has the immediate 1-extension property from M to X if for every x ∉ M, every continuous linear map f : M → Y has a continuous linear extension F : M ⊕ (𝕂x) → Y 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
Definition:[1] A locally convex space Y is injective if for every locally convex space Z containing Y as a topological vector subspace, there exists a continuous projection from Z onto Y.
Definition:[1] A Banach space Y is 1-injective 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

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 Id : Y → Y from Y to the completion Z of Y; i.e. to the map Z → Y).[1]

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

Results

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 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 𝕂. 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 X, then identity map Id : Y → Y can be extended to a continuous linear map of norm 1 to X;
  8. Y is linearly isometric to C(T, 𝕂, ||•||∞) 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 C(T, 𝕂, ||•||∞) for some discrete finite space T.

Examples
Products of the underlying field

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

For any set T, l∞(T) has both the extension property and the metric extension property.

See also

Citations
  1. Narici & Beckenstein 2011, pp. 341–370.

Sources
  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-158488866-6. OCLC 144216834.
  • Rudin, Walter (1 January 1991). Functional Analysis. International Series in Pure and Applied Mathematics (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277 – via Internet Archive.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
  • Trèves, François (6 August 2006) [First published 1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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