Questions tagged [preduals]

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Examples of non $w^{*}$-closed complemented subspaces of a dual Banach space that are also dual spaces

Let $Y$ be a complemented, but not $w^{*}$-closed, subspace of a Banach space $X$. It is known that certain such $Y$ are not dual spaces. Question: What are interesting examples of subspaces of the ...
Jon Bannon's user avatar
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3 votes
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Weak*-separability of the unit ball of $X’$ and density characters and cardinalities of $X$ and $X’$

(This question has also been asked on Math StackExchange.) Let $X$ be a Banach space, $X’$ be its continuous dual such that its unit ball is weak*-separable. I’ve been wondering what can be said about ...
David Gao's user avatar
  • 1,166
2 votes
1 answer

An abstract characterisation of weak* topologies

Is there a way of endowing the unit ball $B_X$ of a Banach space $X$ (we may assume that $X$ is an AL space, if that helps) with a topology $\tau$, so that $\tau=\sigma(Y^*,Y)$ (the weak* topology) if ...
HardyHulley's user avatar
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Predual of $H^{\infty}(\mathbb{D})$

Is the predual of $H^{\infty}(\mathbb{D})$ contained in the maximal ideal space of $H^{\infty}(\mathbb{D})$?
Sherlok's user avatar
  • 149
7 votes
1 answer

Compactness of the unit ball of a Banach space for topologies finer than the weak* topology

Let $(\mathcal{X} , \|\cdot \|_\mathcal{X})$ be a Banach space and $\mathcal{X}'$ its topological dual. We denote by $\| \cdot \|_{\mathcal{X}'}$ the dual norm and define also the topological dual $\...
Goulifet's user avatar
  • 2,152
1 vote
0 answers

Transport-type duality for preduals of $C^{k,1}$-functions

Let $\Omega$ be a non-empty, simply connected, and open subset of $\mathbb{R}^d$ for some positive integer $d$. Let $k$ be a non-negative integer. Consider the Banach space $C^{k,1}_0(\Omega)$ ...
ABIM's user avatar
  • 5,057
3 votes
1 answer

Predual to Lipschitz maps with $p$ derivatives

Let $p\in \mathbb{N}$, and define $\mathrm{Lip}_p$ be the collection of functions from $\mathbb{R}^d$ to itself, with $p-1$ first derivatives bounded and whose $p^{\mathrm{th}}$ derivative is ...
ABIM's user avatar
  • 5,057
1 vote
1 answer

Examples of isomorphic W* algebra with non-homeomorphic weak topology

Due to the uniqueness of the predual, a W* algebra, when realized as a von Neumann algebra in any way, always has a unique, well-defined ultraweak (or $\sigma$-weak) topology. The same can be said ...
Rick Sternbach's user avatar
5 votes
0 answers

On the relation between Lipschitz free-spaces

Let $X$ be a pointed metric space, with base point 0. The space of Lipschitz function which preserves the base point, $Lip_0(X)=\{f:X\to\mathbb{R} : f(0)=0\}$ consider with the norm $\|f(x)\|=\sup_{x\...
Edgaragar's user avatar
5 votes
1 answer

The topological duals of spaces of finite measures

In volume 1 of "Linear Operators", Dunford and Schwartz say that (footnote F1, page 374) "No completely satisfactory representation for the conjugate space of $ba(S, \Sigma)$, $ca(S, \Sigma)$ or $...
Alex M.'s user avatar
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9 votes
3 answers

Lipschitz-free spaces of $\mathbb R^n$

We define $$ \text{Lip}_0(\mathbb R^n)=\{f:\mathbb R^n\rightarrow \mathbb R, \text{such that $f(0)=0$ and } \sup_{x\not=y}\frac{\vert f(x)-f(y)\vert}{\vert x-y\vert}<+\infty. \} $$ It is well-known ...
Bazin's user avatar
  • 15k
3 votes
1 answer

Predual of a subspace

Let $E$ be a Banach space, let $d\ge 1$ be an integer. Let $\mathcal G$ be a weakly closed subspace of $(E^*)^d$ with finite codimension. I would like know if the space $\mathcal G$ is a dual space $\...
Bazin's user avatar
  • 15k
4 votes
1 answer

Predual of a Direct Sum of Banach Spaces

This may be basic, and if it is I apologize, but I have found no references to it in literature. I would appreciate a reference at least if I am wrong. I have supplied background for those interested, ...
Super-Measurable Analyst's user avatar
9 votes
1 answer

Uniqueness up to isometric isomorphism of predual of $(\sum_{\lambda\in\Lambda} H_\lambda)_{l_\infty}$ where $H_\lambda$ are Hilbert spaces

This fact is an easy consequence of results of the paper Classes of Banach spaces with unique isometric preduals. by Leon Brown and Takashi Ito, but it looks like an overkill. Does anyone know a ...
Norbert's user avatar
  • 1,677
11 votes
2 answers

Complexifying a real Banach space and its dual

A standard way to define the "complexification" $E_\mathbb{C}$ of a real Banach space $E$ is to define a complex linear structure on $E\times E$ by (1) $(x,y)+(u,v)=(x+u, y+v)$, (2) $(a+ib)(x,y)=(ax-...
Fred Dashiell's user avatar
1 vote
0 answers

Unambiguous "weak" vector valued $L^{+\infty}$ spaces?

For some time, I have been stuck to the problem to be described as follows. The (perhaps not so commonly known) facts given here are taken from R. E. Edwards' Functional Analysis (Holt, Rinehart and ...
TaQ's user avatar
  • 3,245
6 votes
1 answer

Must a surjective isometry on a dual space have a pre-adjoint?

Background: Let $X$ be a Banach space. We know a linear map $h$ is a surjective isometry of $X$ if and only if its adjoint $h^*$ is a surjective isometry of $X^*$. In general, a linear map $g:X^* \...
user13391's user avatar
21 votes
5 answers

Isomorphisms of Banach Spaces

Suppose $X$ and $Y$ are Banach spaces whose dual spaces are isometrically isomorphic. It is certainly true that $X$ and $Y$ need not be isometrically isomorphic, but must it be true that there is a ...
Mike Hartglass's user avatar