Boundaries, polyhedrality and LFC norms Brazilian Workshop on - PowerPoint PPT Presentation

Boundaries, polyhedrality and LFC norms Brazilian Workshop on geometry of Banach spaces Maresias, 28 August 2014 Richard Smith University College Dublin, Ireland Richard Smith (mathsci.ucd.ie/~rsmith) Boundaries, polyhedrality and LFC norms 28 August 2014 1 / 11

Background Boundaries Boundaries Definition A boundary of a Banach space X is a subset B ⊆ B X ∗ , such that whenever x ∈ X , there exists f ∈ B satisfying f ( x ) = � x � . Richard Smith (mathsci.ucd.ie/~rsmith) Boundaries, polyhedrality and LFC norms 28 August 2014 2 / 11

Background Boundaries Boundaries Definition A boundary of a Banach space X is a subset B ⊆ B X ∗ , such that whenever x ∈ X , there exists f ∈ B satisfying f ( x ) = � x � . Remarks By the Hahn-Banach Theorem, S X ∗ is a boundary. Richard Smith (mathsci.ucd.ie/~rsmith) Boundaries, polyhedrality and LFC norms 28 August 2014 2 / 11

Background Boundaries Boundaries Definition A boundary of a Banach space X is a subset B ⊆ B X ∗ , such that whenever x ∈ X , there exists f ∈ B satisfying f ( x ) = � x � . Remarks By the Hahn-Banach Theorem, S X ∗ is a boundary. By (the proof of) the Krein-Milman Theorem, ext B X ∗ is a boundary. Richard Smith (mathsci.ucd.ie/~rsmith) Boundaries, polyhedrality and LFC norms 28 August 2014 2 / 11

Background Boundaries Boundaries Definition A boundary of a Banach space X is a subset B ⊆ B X ∗ , such that whenever x ∈ X , there exists f ∈ B satisfying f ( x ) = � x � . Remarks By the Hahn-Banach Theorem, S X ∗ is a boundary. By (the proof of) the Krein-Milman Theorem, ext B X ∗ is a boundary. B = {± e ∗ n ∈ ℓ 1 : n ∈ N } is a countable boundary of ( c 0 , �·� ∞ ) . Richard Smith (mathsci.ucd.ie/~rsmith) Boundaries, polyhedrality and LFC norms 28 August 2014 2 / 11

Background Boundaries Boundaries Definition A boundary of a Banach space X is a subset B ⊆ B X ∗ , such that whenever x ∈ X , there exists f ∈ B satisfying f ( x ) = � x � . Remarks By the Hahn-Banach Theorem, S X ∗ is a boundary. By (the proof of) the Krein-Milman Theorem, ext B X ∗ is a boundary. B = {± e ∗ n ∈ ℓ 1 : n ∈ N } is a countable boundary of ( c 0 , �·� ∞ ) . Boundaries can be highly irregular. Richard Smith (mathsci.ucd.ie/~rsmith) Boundaries, polyhedrality and LFC norms 28 August 2014 2 / 11

Background Polyhedral and LFC norms Polyhedral and LFC norms Definition (Klee 60) A norm �·� is polyhedral if, given any finite-dimensional subspace Y ⊆ X , there exist f 1 , . . . , f n ∈ S ( X ∗ , �·� ) such that � y � = max n i = 1 f i ( y ) for all y ∈ Y . Richard Smith (mathsci.ucd.ie/~rsmith) Boundaries, polyhedrality and LFC norms 28 August 2014 3 / 11

Background Polyhedral and LFC norms Polyhedral and LFC norms Definition (Klee 60) A norm �·� is polyhedral if, given any finite-dimensional subspace Y ⊆ X , there exist f 1 , . . . , f n ∈ S ( X ∗ , �·� ) such that � y � = max n i = 1 f i ( y ) for all y ∈ Y . Definition (Pechanec, Whitfield and Zizler 81) A norm �·� depends locally on finitely many coordinates (LFC) if, given 1 x ∈ S X , there exist open U ∋ x and ‘coordinates’ f 1 , . . . , f n ∈ X ∗ , such that Richard Smith (mathsci.ucd.ie/~rsmith) Boundaries, polyhedrality and LFC norms 28 August 2014 3 / 11

Background Polyhedral and LFC norms Polyhedral and LFC norms Definition (Klee 60) A norm �·� is polyhedral if, given any finite-dimensional subspace Y ⊆ X , there exist f 1 , . . . , f n ∈ S ( X ∗ , �·� ) such that � y � = max n i = 1 f i ( y ) for all y ∈ Y . Definition (Pechanec, Whitfield and Zizler 81) A norm �·� depends locally on finitely many coordinates (LFC) if, given 1 x ∈ S X , there exist open U ∋ x and ‘coordinates’ f 1 , . . . , f n ∈ X ∗ , such that � y � = � z � whenever y , z ∈ U and f i ( y ) = f i ( z ) , 1 � i � n . Richard Smith (mathsci.ucd.ie/~rsmith) Boundaries, polyhedrality and LFC norms 28 August 2014 3 / 11

Background Polyhedral and LFC norms Polyhedral and LFC norms Definition (Klee 60) A norm �·� is polyhedral if, given any finite-dimensional subspace Y ⊆ X , there exist f 1 , . . . , f n ∈ S ( X ∗ , �·� ) such that � y � = max n i = 1 f i ( y ) for all y ∈ Y . Definition (Pechanec, Whitfield and Zizler 81) A norm �·� depends locally on finitely many coordinates (LFC) if, given 1 x ∈ S X , there exist open U ∋ x and ‘coordinates’ f 1 , . . . , f n ∈ X ∗ , such that � y � = � z � whenever y , z ∈ U and f i ( y ) = f i ( z ) , 1 � i � n . If all the coordinates come from H ⊆ X ∗ then �·� is LFC- H . 2 Richard Smith (mathsci.ucd.ie/~rsmith) Boundaries, polyhedrality and LFC norms 28 August 2014 3 / 11

Background Polyhedral and LFC norms Polyhedral and LFC norms Definition (Klee 60) A norm �·� is polyhedral if, given any finite-dimensional subspace Y ⊆ X , there exist f 1 , . . . , f n ∈ S ( X ∗ , �·� ) such that � y � = max n i = 1 f i ( y ) for all y ∈ Y . Definition (Pechanec, Whitfield and Zizler 81) A norm �·� depends locally on finitely many coordinates (LFC) if, given 1 x ∈ S X , there exist open U ∋ x and ‘coordinates’ f 1 , . . . , f n ∈ X ∗ , such that � y � = � z � whenever y , z ∈ U and f i ( y ) = f i ( z ) , 1 � i � n . If all the coordinates come from H ⊆ X ∗ then �·� is LFC- H . 2 Example The natural norm on c 0 is both polyhedral and LFC- ( e ∗ n ) n ∈ N . Richard Smith (mathsci.ucd.ie/~rsmith) Boundaries, polyhedrality and LFC norms 28 August 2014 3 / 11

Background The benefits of small boundaries The benefits of small boundaries Theorem (Fonf 89, Hájek 95) Let X be separable. Then the following are equivalent. X has a countable boundary. 1 Richard Smith (mathsci.ucd.ie/~rsmith) Boundaries, polyhedrality and LFC norms 28 August 2014 4 / 11

Background The benefits of small boundaries The benefits of small boundaries Theorem (Fonf 89, Hájek 95) Let X be separable. Then the following are equivalent. X has a countable boundary. 1 X has a norm σ -compact boundary. 2 Richard Smith (mathsci.ucd.ie/~rsmith) Boundaries, polyhedrality and LFC norms 28 August 2014 4 / 11

Background The benefits of small boundaries The benefits of small boundaries Theorem (Fonf 89, Hájek 95) Let X be separable. Then the following are equivalent. X has a countable boundary. 1 X has a norm σ -compact boundary. 2 X has a polyhedral norm (i.e. X is isomorphically polyhedral ). 3 Richard Smith (mathsci.ucd.ie/~rsmith) Boundaries, polyhedrality and LFC norms 28 August 2014 4 / 11

Background The benefits of small boundaries The benefits of small boundaries Theorem (Fonf 89, Hájek 95) Let X be separable. Then the following are equivalent. X has a countable boundary. 1 X has a norm σ -compact boundary. 2 X has a polyhedral norm (i.e. X is isomorphically polyhedral ). 3 X has a polyhedral LFC norm. 4 Richard Smith (mathsci.ucd.ie/~rsmith) Boundaries, polyhedrality and LFC norms 28 August 2014 4 / 11

Background The benefits of small boundaries The benefits of small boundaries Theorem (Fonf 89, Hájek 95) Let X be separable. Then the following are equivalent. X has a countable boundary. 1 X has a norm σ -compact boundary. 2 X has a polyhedral norm (i.e. X is isomorphically polyhedral ). 3 X has a polyhedral LFC norm. 4 X has a LFC norm. 5 Richard Smith (mathsci.ucd.ie/~rsmith) Boundaries, polyhedrality and LFC norms 28 August 2014 4 / 11

Background The benefits of small boundaries The benefits of small boundaries Theorem (Fonf 89, Hájek 95) Let X be separable. Then the following are equivalent. X has a countable boundary. 1 X has a norm σ -compact boundary. 2 X has a polyhedral norm (i.e. X is isomorphically polyhedral ). 3 X has a polyhedral LFC norm. 4 X has a LFC norm. 5 X has a LFC norm that is C ∞ -smooth on X \ { 0 } . 6 Richard Smith (mathsci.ucd.ie/~rsmith) Boundaries, polyhedrality and LFC norms 28 August 2014 4 / 11

Background The benefits of small boundaries The benefits of small boundaries Theorem (Fonf 89, Hájek 95) Let X be separable. Then the following are equivalent. X has a countable boundary. 1 X has a norm σ -compact boundary. 2 X has a polyhedral norm (i.e. X is isomorphically polyhedral ). 3 X has a polyhedral LFC norm. 4 X has a LFC norm. 5 X has a LFC norm that is C ∞ -smooth on X \ { 0 } . 6 Corollaries (Fonf) If X has a norm σ -compact boundary, then. . . Richard Smith (mathsci.ucd.ie/~rsmith) Boundaries, polyhedrality and LFC norms 28 August 2014 4 / 11

Background The benefits of small boundaries The benefits of small boundaries Theorem (Fonf 89, Hájek 95) Let X be separable. Then the following are equivalent. X has a countable boundary. 1 X has a norm σ -compact boundary. 2 X has a polyhedral norm (i.e. X is isomorphically polyhedral ). 3 X has a polyhedral LFC norm. 4 X has a LFC norm. 5 X has a LFC norm that is C ∞ -smooth on X \ { 0 } . 6 Corollaries (Fonf) If X has a norm σ -compact boundary, then. . . X ∗ is separable. Richard Smith (mathsci.ucd.ie/~rsmith) Boundaries, polyhedrality and LFC norms 28 August 2014 4 / 11