The preduals tag has no usage guidance.

**7**

votes

**0**answers

275 views

### 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 ...

**2**

votes

**1**answer

146 views

### 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 $\...

**3**

votes

**1**answer

421 views

### 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, ...

**8**

votes

**1**answer

287 views

### 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 by Leon Brown and Takashi Ito, but it looks like an overkill. Does anyone know a simpler proof?

**9**

votes

**2**answers

841 views

### 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-...

**1**

vote

**0**answers

287 views

### 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 ...

**6**

votes

**1**answer

556 views

### 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^* \...

**18**

votes

**5**answers

2k views

### 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 ...