## New answers tagged banach-spaces

6
votes

### Bilipschitz embedding of the unit ball of $c_0$ into $\ell_1$

Yes, this is known. Raynaud showed that $B_{c_0}$ does not uniformly (in particular, bilipschitz) embed into any stable Banach space. $\ell_1$ is stable.
Yves Raynaud, Espaces de Banach superstables, ...

- 2,104

0
votes

Accepted

### Finding the set of best approximation

Similar to $P_Y(x)$, there is no such ready formula for evaluating $P_{B_Y}(x)$, when $Y=\ker (f)$, and so is for $d(x,B_Y)$. In some cases, for instance when $d(x,Y)=d(x,B_Y)$, it is easier to ...

- 541

8
votes

Accepted

### Banach space with uncountable basis

I will pull together the comments into a community wiki answer with some of my own remarks so that the question isn't left on the unanswered questions list.
If you're willing to accept that it is ...

Community wiki

1
vote

Accepted

### Does $L_{p}(\mu, X)^*=L_{q} (\mu, X^*)$ hold for $\sigma$-finite measure spaces?

Below is my formalization of @Nik's hints to finish the proof.
Let's prove that
$$
\sum_{m=1}^M \|H_m\|^q_{L_{p}(\mu_m, X)^*} \le \|H\|^q_{L_{p}(\mu, X)^*} \quad \forall M \in \mathbb N^*.
$$
Let $\...

- 347

3
votes

Accepted

### If $H \in L_{q} (\mu, X^*)$ such that $\int \langle H, f \rangle \mathrm d \mu = 0$ for all $f \in L_{p}(\mu, X)$, then $H=0$ $\mu$-a.e

It is always true that $L_q(\mu,X^*)\hookrightarrow L_p(\mu,X)^*$ isometrically (without the assumption that $X$ has the Radon-Nikodym property). We need the Radon-Nikodym property to guarantee that ...

- 91

2
votes

Accepted

### Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace - Part II

Gro-Tsen's answer to your previous question provides a counterexample if you define $D$ to be all vectors in $\ell_2$ that are of the form $\sum_n a_n f_n$, where
$f_n = e_n + e_{n+1}$, $(e_n)$ is the ...

- 30k

1
vote

### On the measure of nonconvexity (MNC)

This is a simple problem and I am voting to close it. Take any non-convex set $A$ with $\alpha(A)<\infty$ and define $A_n=\lambda^n A$ (dilation with the factor $\lambda^n$. Then $\alpha(A_n)=\...

- 24k

0
votes

Accepted

### Differentiability of the fixed points of a family of contraction maps

I found the answer myself: One can simply apply the Banach space version of the implicit function theorem to the function $G(t,x) = x-F_t(x)$. The implicit function theorem shows that, given $G$ is in ...

- 121

14
votes

Accepted

### Converse of closed graph theorem

No. The closed graph theorem in this form is equivalent to $X$ being a barreled space. See item 15 here.
There are incomplete normed spaces that are barreled. See here.

- 17.5k

3
votes

Accepted

### Complemented subspaces in a dual Banach space

$L^1$ is complemented in the measure space $M([0,1])$, $L^1$ is not a dual space.

- 1,443

0
votes

Accepted

### A generalization about the density of $\mathcal C_c(X, E)$ in $\mathcal L_p (X, \mu, E)$ when $E$ is a Banach space

Below is a counter-example taken from this thread. It works even when $\mathcal C_c$ is replaced by $\mathcal C$, the space of all continuous functions from $X$ from $E$.
Let $X:=[0, 1]$, $E:=\mathbb ...

- 349

2
votes

Accepted

### Is $L_p(X, \mu, E)$ uniformly convex for $p \in (1, \infty)$ if $E$ is a uniformly convex Banach space?

A reference for this result would be Some more uniformly convex spaces by Mahlon M. Day, Bull. Amer. Math. Soc. 47(6): 504-507 (June 1941).
(Alternative link at Project Euclid)

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