# Tag Info

## New answers tagged banach-spaces

### 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, ...
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### 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 ...
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### 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
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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 ...
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### 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.
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### 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.
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