Embedding Banach Spaces

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Embedding Banach spaces.

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, and Andras Zsak

January 24, 2009

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Deﬁnition (Banach space) A normed vector space (X , k · k) is called a Banach space if X is complete in the norm topology.

Deﬁnition (Schauder basis) ∞ A sequence (xi )i=1 ⊂ X is called a basis for X if for every x ∈ X there exists a ∞ P∞ unique sequence of scalars (ai )i=1 such that x = i=1 ai xi .

Question (Mazur ’36) Does every separable Banach space have a basis?

Theorem (Enﬂo ’72) No.

Background

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Deﬁnition (Schauder basis) ∞ A sequence (xi )i=1 ⊂ X is called a basis for X if for every x ∈ X there exists a ∞ P∞ unique sequence of scalars (ai )i=1 such that x = i=1 ai xi .

Question (Mazur ’36) Does every separable Banach space have a basis?

Theorem (Enﬂo ’72) No.

Background

Deﬁnition (Banach space) A normed vector space (X , k · k) is called a Banach space if X is complete in the norm topology.

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Question (Mazur ’36) Does every separable Banach space have a basis?

Theorem (Enﬂo ’72) No.

Background

Deﬁnition (Banach space) A normed vector space (X , k · k) is called a Banach space if X is complete in the norm topology.

Deﬁnition (Schauder basis) ∞ A sequence (xi )i=1 ⊂ X is called a basis for X if for every x ∈ X there exists a ∞ P∞ unique sequence of scalars (ai )i=1 such that x = i=1 ai xi .

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Theorem (Enﬂo ’72) No.

Background

Deﬁnition (Banach space) A normed vector space (X , k · k) is called a Banach space if X is complete in the norm topology.

Deﬁnition (Schauder basis) ∞ A sequence (xi )i=1 ⊂ X is called a basis for X if for every x ∈ X there exists a ∞ P∞ unique sequence of scalars (ai )i=1 such that x = i=1 ai xi .

Question (Mazur ’36) Does every separable Banach space have a basis?

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. No.

Background

Deﬁnition (Banach space) A normed vector space (X , k · k) is called a Banach space if X is complete in the norm topology.

Deﬁnition (Schauder basis) ∞ A sequence (xi )i=1 ⊂ X is called a basis for X if for every x ∈ X there exists a ∞ P∞ unique sequence of scalars (ai )i=1 such that x = i=1 ai xi .

Question (Mazur ’36) Does every separable Banach space have a basis?

Theorem (Enﬂo ’72)

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Background

Deﬁnition (Banach space) A normed vector space (X , k · k) is called a Banach space if X is complete in the norm topology.

Deﬁnition (Schauder basis) ∞ A sequence (xi )i=1 ⊂ X is called a basis for X if for every x ∈ X there exists a ∞ P∞ unique sequence of scalars (ai )i=1 such that x = i=1 ai xi .

Question (Mazur ’36) Does every separable Banach space have a basis?

Theorem (Enﬂo ’72) No.

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Every separable Banach space is isometric to a subspace of C[0, 1].

Question Given a Banach space X with some desirable structure, does X embed into a Banach space Y with a basis having some related structure?

Theorem (Zippin ’88) A separable Banach space X is reﬂexive if and only if X embeds into a Banach space Y with a basis which is boundedly complete and shrinking. A separable Banach space X has separable dual if and only if X embeds into a Banach space Y with a basis which is shrinking.

Old embedding results

Theorem (Banach)

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Question Given a Banach space X with some desirable structure, does X embed into a Banach space Y with a basis having some related structure?

Old embedding results

Theorem (Banach) Every separable Banach space is isometric to a subspace of C[0, 1].

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Theorem (Zippin ’88) A separable Banach space X is reﬂexive if and only if X embeds into a Banach space Y with a basis which is boundedly complete and shrinking. A separable Banach space X has separable dual if and only if X embeds into a Banach space Y with a basis which is shrinking.

Old embedding results

Theorem (Banach) Every separable Banach space is isometric to a subspace of C[0, 1].

Question Given a Banach space X with some desirable structure, does X embed into a Banach space Y with a basis having some related structure?

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. A separable Banach space X is reﬂexive if and only if X embeds into a Banach space Y with a basis which is boundedly complete and shrinking. A separable Banach space X has separable dual if and only if X embeds into a Banach space Y with a basis which is shrinking.

Old embedding results

Theorem (Banach) Every separable Banach space is isometric to a subspace of C[0, 1].

Question Given a Banach space X with some desirable structure, does X embed into a Banach space Y with a basis having some related structure?

Theorem (Zippin ’88)

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. A separable Banach space X has separable dual if and only if X embeds into a Banach space Y with a basis which is shrinking.

Old embedding results

Theorem (Banach) Every separable Banach space is isometric to a subspace of C[0, 1].

Question Given a Banach space X with some desirable structure, does X embed into a Banach space Y with a basis having some related structure?

Theorem (Zippin ’88) A separable Banach space X is reﬂexive if and only if X embeds into a Banach space Y with a basis which is boundedly complete and shrinking.

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Old embedding results

Theorem (Banach) Every separable Banach space is isometric to a subspace of C[0, 1].

Question Given a Banach space X with some desirable structure, does X embed into a Banach space Y with a basis having some related structure?

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Deﬁnition (Schauder frame) ∞ Let X be an inﬁnite dimensional separable Banach space. A sequence (xi , fi )i=1 ∞ ∞ ∗ with (xi )i=1 ⊂ X and (fi )i=1 ⊂ X is called a frame of X if for every x ∈ X , P x = fi (x)xi . ∞ P The sequence (xi , fi )i=1 is called an unconditional frame of X if fi (x)xi converges unconditionally to x ∈ X .

Theorem (Casazza, Han, Larson ’99 and Casazza, Dilworth, Odell, Schlumprecht, Zsak ’07) ∞ ∞ ∗ Let X be a separable Banach space and let (xi )i=1 ⊂ X and (fi )i=1 ⊂ X . ∞ (xi , fi )i=1 is a frame of X if and only if there is a Banach space Z with a basis ∞ ∗ ∞ (zi )i=1 having coordinate functionals (zi )i=1 and an isomorphic embedding

T : X → Z and a bounded linear surjection S : Z → X such that S ◦ T = IdX ∗ ∗ and S(zi ) = xi and T (zi ) = fi for all i ∈ N with xi 6= 0.

Essentially, frames are characterized as projections of bases onto complemented subspaces.

Banach frames

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. ∞ Let X be an inﬁnite dimensional separable Banach space. A sequence (xi , fi )i=1 ∞ ∞ ∗ with (xi )i=1 ⊂ X and (fi )i=1 ⊂ X is called a frame of X if for every x ∈ X , P x = fi (x)xi . ∞ P The sequence (xi , fi )i=1 is called an unconditional frame of X if fi (x)xi converges unconditionally to x ∈ X .

T : X → Z and a bounded linear surjection S : Z → X such that S ◦ T = IdX ∗ ∗ and S(zi ) = xi and T (zi ) = fi for all i ∈ N with xi 6= 0.

Essentially, frames are characterized as projections of bases onto complemented subspaces.

Banach frames

Deﬁnition (Schauder frame)

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. ∞ P The sequence (xi , fi )i=1 is called an unconditional frame of X if fi (x)xi converges unconditionally to x ∈ X .

T : X → Z and a bounded linear surjection S : Z → X such that S ◦ T = IdX ∗ ∗ and S(zi ) = xi and T (zi ) = fi for all i ∈ N with xi 6= 0.

Essentially, frames are characterized as projections of bases onto complemented subspaces.

Banach frames

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Theorem (Casazza, Han, Larson ’99 and Casazza, Dilworth, Odell, Schlumprecht, Zsak ’07) ∞ ∞ ∗ Let X be a separable Banach space and let (xi )i=1 ⊂ X and (fi )i=1 ⊂ X . ∞ (xi , fi )i=1 is a frame of X if and only if there is a Banach space Z with a basis ∞ ∗ ∞ (zi )i=1 having coordinate functionals (zi )i=1 and an isomorphic embedding

T : X → Z and a bounded linear surjection S : Z → X such that S ◦ T = IdX ∗ ∗ and S(zi ) = xi and T (zi ) = fi for all i ∈ N with xi 6= 0.

Essentially, frames are characterized as projections of bases onto complemented subspaces.

Banach frames

Deﬁnition (Schauder frame) ∞ Let X be an inﬁnite dimensional separable Banach space. A sequence (xi , fi )i=1 ∞ ∞ ∗ with (xi )i=1 ⊂ X and (fi )i=1 ⊂ X is called a frame of X if for every x ∈ X , P x = fi (x)xi . ∞ P The sequence (xi , fi )i=1 is called an unconditional frame of X if fi (x)xi converges unconditionally to x ∈ X .

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. ∞ ∞ ∗ Let X be a separable Banach space and let (xi )i=1 ⊂ X and (fi )i=1 ⊂ X . ∞ (xi , fi )i=1 is a frame of X if and only if there is a Banach space Z with a basis ∞ ∗ ∞ (zi )i=1 having coordinate functionals (zi )i=1 and an isomorphic embedding

T : X → Z and a bounded linear surjection S : Z → X such that S ◦ T = IdX ∗ ∗ and S(zi ) = xi and T (zi ) = fi for all i ∈ N with xi 6= 0.

Essentially, frames are characterized as projections of bases onto complemented subspaces.

Banach frames

Theorem (Casazza, Han, Larson ’99 and Casazza, Dilworth, Odell, Schlumprecht, Zsak ’07)

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. ∞ (xi , fi )i=1 is a frame of X if and only if there is a Banach space Z with a basis ∞ ∗ ∞ (zi )i=1 having coordinate functionals (zi )i=1 and an isomorphic embedding

T : X → Z and a bounded linear surjection S : Z → X such that S ◦ T = IdX ∗ ∗ and S(zi ) = xi and T (zi ) = fi for all i ∈ N with xi 6= 0.

Essentially, frames are characterized as projections of bases onto complemented subspaces.

Banach frames

Theorem (Casazza, Han, Larson ’99 and Casazza, Dilworth, Odell, Schlumprecht, Zsak ’07) ∞ ∞ ∗ Let X be a separable Banach space and let (xi )i=1 ⊂ X and (fi )i=1 ⊂ X .

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Essentially, frames are characterized as projections of bases onto complemented subspaces.

Banach frames

T : X → Z and a bounded linear surjection S : Z → X such that S ◦ T = IdX ∗ ∗ and S(zi ) = xi and T (zi ) = fi for all i ∈ N with xi 6= 0.

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Banach frames

T : X → Z and a bounded linear surjection S : Z → X such that S ◦ T = IdX ∗ ∗ and S(zi ) = xi and T (zi ) = fi for all i ∈ N with xi 6= 0.

Essentially, frames are characterized as projections of bases onto complemented subspaces. Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Theorem (Pelczynski ’71 Johnson, Rosenthal and Zippin ’71) A Banach space X has the bounded approximation property if and only if X is isomorphic to a complemented subspace of a Banach space with a basis.

Corollary A Banach space X has a frame if and only if X has the bounded approximation property.

Deﬁnition (Bounded approximation property) A Banach space X has the bounded approximation property if there exists a constant λ ≥ 1 such that for all > 0 and compact K ⊂ X there exists a ﬁnite rank operator T : X → X with kT k ≤ λ and kT (x) − xk < ∀x ∈ X .

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Corollary A Banach space X has a frame if and only if X has the bounded approximation property.

Theorem (Pelczynski ’71 Johnson, Rosenthal and Zippin ’71) A Banach space X has the bounded approximation property if and only if X is isomorphic to a complemented subspace of a Banach space with a basis.

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Deﬁnition (Bounded approximation property) A Banach space X has the bounded approximation property if there exists a constant λ ≥ 1 such that for all > 0 and compact K ⊂ X there exists a ﬁnite rank operator T : X → X with kT k ≤ λ and kT (x) − xk < ∀x ∈ X .

Corollary A Banach space X has a frame if and only if X has the bounded approximation property.

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. In 2002, Odell and Schlumprecht characterize when a separable reﬂexive

Banach space embeds into a Banach space of the form Y = (⊕En)`p where En is a ﬁnite dimensional space for all n ∈ N. In fact, Y can be chosen to be n Y = (⊕`∞)`p . In 2008, Johnson and Zheng characterized when a separable reﬂexive Banach space embeds into a Banach space with an unconditional basis.

Question (70’s) Does every Banach space with an unconditional frame have an unconditional basis?

Recent embedding results

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. In fact, Y can be chosen to be n Y = (⊕`∞)`p . In 2008, Johnson and Zheng characterized when a separable reﬂexive Banach space embeds into a Banach space with an unconditional basis.

Question (70’s) Does every Banach space with an unconditional frame have an unconditional basis?

Recent embedding results

In 2002, Odell and Schlumprecht characterize when a separable reﬂexive

Banach space embeds into a Banach space of the form Y = (⊕En)`p where En is a ﬁnite dimensional space for all n ∈ N.

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. In 2008, Johnson and Zheng characterized when a separable reﬂexive Banach space embeds into a Banach space with an unconditional basis.

Question (70’s) Does every Banach space with an unconditional frame have an unconditional basis?

Recent embedding results

In 2002, Odell and Schlumprecht characterize when a separable reﬂexive

Banach space embeds into a Banach space of the form Y = (⊕En)`p where En is a ﬁnite dimensional space for all n ∈ N. In fact, Y can be chosen to be n Y = (⊕`∞)`p .

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Question (70’s) Does every Banach space with an unconditional frame have an unconditional basis?

Recent embedding results

In 2002, Odell and Schlumprecht characterize when a separable reﬂexive

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Recent embedding results

In 2002, Odell and Schlumprecht characterize when a separable reﬂexive

Question (70’s) Does every Banach space with an unconditional frame have an unconditional basis?

Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Theorem (Szlenk ’68) Every separable Banach space X may be assigned an ordinal Sz(X ) such that 1. If X ⊂ Y then Sz(X ) ≤ Sz(Y ). 2. X has separable dual if and only if Sz(X ) is countable. 3. For every countable ordinal α there exists a Banach space X with separable dual such that Sz(X ) > α. In particular, there does not exist a Banach space Y with separable dual such that every separable reﬂexive Banach space embeds into Y .

Question (Bourgain ’80) Does there exist a separable reﬂexive Banach space Y such that every separable uniformly convex Banach space embeds into Y ?

Theorem (Odell, Schlumprecht ’07) Yes.