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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, ...
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 ...
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 ...
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 ...
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 ...
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)=\...
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.
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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