On some Banach spaces of martingales associated with function spaces

We consider pairs of Banach spaces $(M_0, M)$ such that $M_0$ is defined in terms of a little-$o$ condition, and $M$ is defined by the corresponding big-$O$ condition. The construction is general and pairs include function spaces of vanishing and bounded mean oscillation, vanishing weighted and weighted spaces of functions or their derivatives, Möbius invariant spaces of analytic functions, Lipschitz-Hölder spaces, etc. It has previously been shown that the bidual $M_0^{**}$ of $M_0$ is isometrically isomorphic with $M$. The main result of this paper is that $M_0$ is an M-ideal in $M$. This has several useful consequences: $M_0$ has Pełczýnskis properties (u) and (V), $M_0$ is proximinal in $M$, and $M_0^*$ is a strongly unique predual of $M$, while $M_0$ itself never is a strongly unique predual.

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Function Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-1953-031-3 ◽

1953 ◽

Vol 5 ◽

pp. 273-288 ◽

Cited By ~ 43

Author(s):

Israel Halperin

Keyword(s):

Banach Spaces ◽

Measure Space ◽

Function Spaces ◽

Weight Functions ◽

Negative Weight ◽

Image Position

This paper is the first in a series dealing with Banach spaces L whose elements are functions on a measure space S. If W is a family of non-negative weight functions wα we sometimes write LWp when the norm is given as

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Function Spaces and Banach Spaces

Real and Abstract Analysis ◽

10.1007/978-3-642-88047-6_4 ◽

1965 ◽

pp. 188-255

Author(s):

Edwin Hewitt ◽

Karl Stromberg

Keyword(s):

Banach Spaces ◽

Function Spaces

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Function Spaces and Banach Spaces

Real and Abstract Analysis ◽

10.1007/978-3-642-88044-5_4 ◽

1965 ◽

pp. 188-255

Author(s):

Edwin Hewitt ◽

Karl Stromberg

Keyword(s):

Banach Spaces ◽

Function Spaces

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New Sequence Spaces and Function Spaces on Interval[0,1]

Journal of Function Spaces and Applications ◽

10.1155/2013/601490 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10

Author(s):

Cheng-Zhong Xu ◽

Gen-Qi Xu

Keyword(s):

Banach Spaces ◽

Function Spaces ◽

Summation Method ◽

Sequence Spaces ◽

Polynomial Space ◽

And Function

We study the sequence spaces and the spaces of functions defined on interval0,1in this paper. By a new summation method of sequences, we find out some new sequence spaces that are interpolating into spaces betweenℓpandℓqand function spaces that are interpolating into the spaces between the polynomial spaceP0,1andC∞0,1. We prove that these spaces of sequences and functions are Banach spaces.

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Kadec-Klee property in Musielak-Orlicz function spaces equipped with the Orlicz norm

Aequationes Mathematicae ◽

10.1007/s00010-021-00808-8 ◽

2021 ◽

Author(s):

Yunan Cui ◽

Li Zhao

Keyword(s):

Fixed Point ◽

Banach Spaces ◽

Orlicz Function ◽

Function Spaces ◽

Fixed Point Property ◽

Point Property ◽

Orlicz Norm

AbstractIt is well-known that the Kadec-Klee property is an important property in the geometry of Banach spaces. It is closely connected with the approximation compactness and fixed point property of non-expansive mappings. In this paper, a criterion for Musielak-Orlicz function spaces equipped with the Orlicz norm to have the Kadec-Klee property are given. As a corollary, we obtain that a class of non-reflexive Musielak-Orlicz function spaces have the Fixed Point property.

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Multiresolution analysis for compactly supported interpolating tensor product wavelets

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691315500101 ◽

2015 ◽

Vol 13 (02) ◽

pp. 1550010 ◽

Cited By ~ 1

Author(s):

Tommi Höynälänmaa

Keyword(s):

Tensor Product ◽

Banach Spaces ◽

Multiresolution Analysis ◽

Continuous Functions ◽

Function Spaces ◽

One Dimensional ◽

Space Norm ◽

The One ◽

Complex Valued ◽

Uniformly Continuous

We construct multidimensional interpolating tensor product multiresolution analyses (MRA's) of the function spaces C0(ℝn, K), K = ℝ or K = ℂ, consisting of real or complex valued functions on ℝn vanishing at infinity and the function spaces Cu(ℝn, K) consisting of bounded and uniformly continuous functions on ℝn. We also construct an interpolating dual MRA for both of these spaces. The theory of the tensor products of Banach spaces is used. We generalize the Besov space norm equivalence from the one-dimensional case to our n-dimensional construction.

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