Multinorms and Banach lattices Based on results of G.Dales, - PowerPoint PPT Presentation

Multinorms and Banach lattices Based on results of G.Dales, M.Polyakov, N.Laustsen, G.Pisier, L.McClaran, P.Ramsden, T.Oikhberg Vladimir Troitsky University of Alberta July 2014

Multinorms

Multinorms Given a vector space X .

Multinorms Given a vector space X . For each n , given a norm �·� n on X n such that

Multinorms Given a vector space X . For each n , given a norm �·� n on X n such that � � � � (A1) � ( x σ (1) , . . . , x σ ( n ) ) n = � ( x 1 , . . . , x n ) � � n

Multinorms Given a vector space X . For each n , given a norm �·� n on X n such that � � � � (A1) � ( x σ (1) , . . . , x σ ( n ) ) n = � ( x 1 , . . . , x n ) � � n � � � � (A2) � ( x 1 , . . . , x n , 0) n +1 = � ( x 1 , . . . , x n ) � � n

Multinorms Given a vector space X . For each n , given a norm �·� n on X n such that � � � � (A1) � ( x σ (1) , . . . , x σ ( n ) ) n = � ( x 1 , . . . , x n ) � � n � � � � (A2) � ( x 1 , . . . , x n , 0) n +1 = � ( x 1 , . . . , x n ) � � n � � � � (A3) � ( α 1 x 1 , . . . , α n x n ) n � max | α i | · � ( x 1 , . . . , x n ) � � n

Multinorms Given a vector space X . For each n , given a norm �·� n on X n such that � � � � (A1) � ( x σ (1) , . . . , x σ ( n ) ) n = � ( x 1 , . . . , x n ) � � n � � � � (A2) � ( x 1 , . . . , x n , 0) n +1 = � ( x 1 , . . . , x n ) � � n � � � � (A3) � ( α 1 x 1 , . . . , α n x n ) n � max | α i | · � ( x 1 , . . . , x n ) � � n � � � � (A4) � ( x 1 , . . . , x n − 1 , x n , x n ) n +1 = � ( x 1 , . . . , x n − 1 , x n ) � � n

Multinorms Given a vector space X . For each n , given a norm �·� n on X n such that � � � � (A1) � ( x σ (1) , . . . , x σ ( n ) ) n = � ( x 1 , . . . , x n ) � � n � � � � (A2) � ( x 1 , . . . , x n , 0) n +1 = � ( x 1 , . . . , x n ) � � n � � � � (A3) � ( α 1 x 1 , . . . , α n x n ) n � max | α i | · � ( x 1 , . . . , x n ) � � n � � � � (A4) � ( x 1 , . . . , x n − 1 , x n , x n ) n +1 = � ( x 1 , . . . , x n − 1 , x n ) � � n Such a sequence of norms is called a multinorm on X .

Multinorms Given a vector space X . For each n , given a norm �·� n on X n such that � � � � (A1) � ( x σ (1) , . . . , x σ ( n ) ) n = � ( x 1 , . . . , x n ) � � n � � � � (A2) � ( x 1 , . . . , x n , 0) n +1 = � ( x 1 , . . . , x n ) � � n � � � � (A3) � ( α 1 x 1 , . . . , α n x n ) n � max | α i | · � ( x 1 , . . . , x n ) � � n � � � � (A4) � ( x 1 , . . . , x n − 1 , x n , x n ) n +1 = � ( x 1 , . . . , x n − 1 , x n ) � � n Such a sequence of norms is called a multinorm on X . Example � := max � x i � . � � Let X be a normed space. Put � ( x 1 , . . . , x n )

Multinorms Given a vector space X . For each n , given a norm �·� n on X n such that � � � � (A1) � ( x σ (1) , . . . , x σ ( n ) ) n = � ( x 1 , . . . , x n ) � � n � � � � (A2) � ( x 1 , . . . , x n , 0) n +1 = � ( x 1 , . . . , x n ) � � n � � � � (A3) � ( α 1 x 1 , . . . , α n x n ) n � max | α i | · � ( x 1 , . . . , x n ) � � n � � � � (A4) � ( x 1 , . . . , x n − 1 , x n , x n ) n +1 = � ( x 1 , . . . , x n − 1 , x n ) � � n Such a sequence of norms is called a multinorm on X . Example � := max � x i � . � � Let X be a normed space. Put � ( x 1 , . . . , x n ) Example � � � := � n � � Let X be a Banach lattice. Put � ( x 1 , . . . , x n ) i =1 | x i | � . � � �

Multinorms Given a vector space X . For each n , given a norm �·� n on X n such that � � � � (A1) � ( x σ (1) , . . . , x σ ( n ) ) n = � ( x 1 , . . . , x n ) � � n � � � � (A2) � ( x 1 , . . . , x n , 0) n +1 = � ( x 1 , . . . , x n ) � � n � � � � (A3) � ( α 1 x 1 , . . . , α n x n ) n � max | α i | · � ( x 1 , . . . , x n ) � � n � � � � (A4) � ( x 1 , . . . , x n − 1 , x n , x n ) n +1 = � ( x 1 , . . . , x n − 1 , x n ) � � n Such a sequence of norms is called a multinorm on X . Example � := max � x i � . � � Let X be a normed space. Put � ( x 1 , . . . , x n ) Example � � � := � n � � Let X be a Banach lattice. Put � ( x 1 , . . . , x n ) i =1 | x i | � . � � � The only multinorm on R is the ℓ ∞ -norm.

1-multinorms A sequence of norms on X n is a 1-multimorm if it satisfies � � � � (A1) � ( x σ (1) , . . . , x σ ( n ) ) n = � ( x 1 , . . . , x n ) � � n � � � � (A2) � ( x 1 , . . . , x n , 0) n +1 = � ( x 1 , . . . , x n ) � � n � � � � (A3) � ( α 1 x 1 , . . . , α n x n ) n � max | α i | · � ( x 1 , . . . , x n ) � � n � � � � (A4’) � ( x 1 , . . . , x n − 1 , x n , x n ) n +1 = � ( x 1 , . . . , x n − 1 , 2 x n ) n . � �

1-multinorms A sequence of norms on X n is a 1-multimorm if it satisfies � � � � (A1) � ( x σ (1) , . . . , x σ ( n ) ) n = � ( x 1 , . . . , x n ) � � n � � � � (A2) � ( x 1 , . . . , x n , 0) n +1 = � ( x 1 , . . . , x n ) � � n � � � � (A3) � ( α 1 x 1 , . . . , α n x n ) n � max | α i | · � ( x 1 , . . . , x n ) � � n � � � � (A4’) � ( x 1 , . . . , x n − 1 , x n , x n ) n +1 = � ( x 1 , . . . , x n − 1 , 2 x n ) n . � � Example � := � n � � Let X be a normed space. Put � ( x 1 , . . . , x n ) i =1 � x i � .

1-multinorms A sequence of norms on X n is a 1-multimorm if it satisfies � � � � (A1) � ( x σ (1) , . . . , x σ ( n ) ) n = � ( x 1 , . . . , x n ) � � n � � � � (A2) � ( x 1 , . . . , x n , 0) n +1 = � ( x 1 , . . . , x n ) � � n � � � � (A3) � ( α 1 x 1 , . . . , α n x n ) n � max | α i | · � ( x 1 , . . . , x n ) � � n � � � � (A4’) � ( x 1 , . . . , x n − 1 , x n , x n ) n +1 = � ( x 1 , . . . , x n − 1 , 2 x n ) n . � � Example � := � n � � Let X be a normed space. Put � ( x 1 , . . . , x n ) i =1 � x i � . Example � � � := � n � � Let X be a Banach lattice. Put � ( x 1 , . . . , x n ) i =1 | x i | � . � � �

1-multinorms A sequence of norms on X n is a 1-multimorm if it satisfies � � � � (A1) � ( x σ (1) , . . . , x σ ( n ) ) n = � ( x 1 , . . . , x n ) � � n � � � � (A2) � ( x 1 , . . . , x n , 0) n +1 = � ( x 1 , . . . , x n ) � � n � � � � (A3) � ( α 1 x 1 , . . . , α n x n ) n � max | α i | · � ( x 1 , . . . , x n ) � � n � � � � (A4’) � ( x 1 , . . . , x n − 1 , x n , x n ) n +1 = � ( x 1 , . . . , x n − 1 , 2 x n ) n . � � Example � := � n � � Let X be a normed space. Put � ( x 1 , . . . , x n ) i =1 � x i � . Example � � � := � n � � Let X be a Banach lattice. Put � ( x 1 , . . . , x n ) i =1 | x i | � . � � � The only 1-multinorm on R is the ℓ 1 -norm.

Theorem A sequence of norms is a multinorm iff x ∈ X n and every A ∈ M m , n ; � A ¯ x � m � � A �� ¯ x � n for every ¯

Theorem A sequence of norms is a multinorm iff x ∈ X n and every A ∈ M m , n ; � A ¯ x � m � � A �� ¯ x � n for every ¯ x ) i = � n where ( A ¯ j =1 a ij x j

Theorem A sequence of norms is a multinorm iff x ∈ X n and every A ∈ M m , n ; � A ¯ x � m � � A �� ¯ x � n for every ¯ x ) i = � n j =1 a ij x j and � A � = � A : ℓ n ∞ → ℓ m where ( A ¯ ∞ �

Theorem A sequence of norms is a multinorm iff x ∈ X n and every A ∈ M m , n ; � A ¯ x � m � � A �� ¯ x � n for every ¯ x ) i = � n j =1 a ij x j and � A � = � A : ℓ n ∞ → ℓ m where ( A ¯ ∞ � Theorem A sequence of norms is a 1-multinorm iff x ∈ X n and A ∈ M m , n . x � m � � A : ℓ n 1 → ℓ m � A ¯ 1 � · � ¯ x � n for every ¯

Theorem A sequence of norms is a multinorm iff x ∈ X n and every A ∈ M m , n ; � A ¯ x � m � � A �� ¯ x � n for every ¯ x ) i = � n j =1 a ij x j and � A � = � A : ℓ n ∞ → ℓ m where ( A ¯ ∞ � Theorem A sequence of norms is a 1-multinorm iff x ∈ X n and A ∈ M m , n . x � m � � A : ℓ n 1 → ℓ m � A ¯ 1 � · � ¯ x � n for every ¯ Definition Given 1 � p � ∞ , we say that a sequence of norms on X n is a x � m � � A : ℓ n p → ℓ m x ∈ X n p -multinorm if � A ¯ p � · � ¯ x � n for every ¯ and A ∈ M m , n .