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) 1 A sequence (xi )i=1 ⊂ X is called a basis for X if for every x 2 X there exists a 1 P1 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) 1 A sequence (xi )i=1 ⊂ X is called a basis for X if for every x 2 X there exists a 1 P1 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) 1 A sequence (xi )i=1 ⊂ X is called a basis for X if for every x 2 X there exists a 1 P1 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) 1 A sequence (xi )i=1 ⊂ X is called a basis for X if for every x 2 X there exists a 1 P1 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) 1 A sequence (xi )i=1 ⊂ X is called a basis for X if for every x 2 X there exists a 1 P1 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) 1 A sequence (xi )i=1 ⊂ X is called a basis for X if for every x 2 X there exists a 1 P1 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? 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]. 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? 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. Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Definition (Schauder frame) 1 Let X be an infinite dimensional separable Banach space. A sequence (xi ; fi )i=1 1 1 ∗ with (xi )i=1 ⊂ X and (fi )i=1 ⊂ X is called a frame of X if for every x 2 X , P x = fi (x)xi . 1 P The sequence (xi ; fi )i=1 is called an unconditional frame of X if fi (x)xi converges unconditionally to x 2 X . Theorem (Casazza, Han, Larson '99 and Casazza, Dilworth, Odell, Schlumprecht, Zsak '07) 1 1 ∗ Let X be a separable Banach space and let (xi )i=1 ⊂ X and (fi )i=1 ⊂ X . 1 (xi ; fi )i=1 is a frame of X if and only if there is a Banach space Z with a basis 1 ∗ 1 (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 2 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. 1 Let X be an infinite dimensional separable Banach space. A sequence (xi ; fi )i=1 1 1 ∗ with (xi )i=1 ⊂ X and (fi )i=1 ⊂ X is called a frame of X if for every x 2 X , P x = fi (x)xi . 1 P The sequence (xi ; fi )i=1 is called an unconditional frame of X if fi (x)xi converges unconditionally to x 2 X . Theorem (Casazza, Han, Larson '99 and Casazza, Dilworth, Odell, Schlumprecht, Zsak '07) 1 1 ∗ Let X be a separable Banach space and let (xi )i=1 ⊂ X and (fi )i=1 ⊂ X . 1 (xi ; fi )i=1 is a frame of X if and only if there is a Banach space Z with a basis 1 ∗ 1 (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 2 N with xi 6= 0.