# Tagged Questions

A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

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### Image of a set under a mapping

I need to show that the image of the closed unit ball in $\mathbb{C}$, under the polynomial mapping $p(x) = (1-x)^2$ is the cardioid: ${re^{i\theta} : 0 \leq \theta < 2π, 0 ≤ r ≤ 2 + 2 \cos\theta}$...
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If ${c_n}$ is a bounded sequence, then $$f(r, \theta)=\sum_{n=-\infty}^{\infty}c_nr^{|n|}e^{in\theta}$$ is harmonic in the disc. Help me proving?
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### separable Banach space with Banach-Mazur distances to $\ell_2^n$ bounded must be isomorphic to $\ell_2$?

If $X$ is a separable infinite-dimensional Banach space and $C\in\mathbb{R}^+$ is an upper bound for the Banach-Mazur distance $d(E,\ell_2^n)$ for all $n\in\mathbb{N}$ and all $n$-dimensional $E\leq X$...
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### existence of invertible operator mapping one sequence pointwise to a 'nearby' sequence

Let $X$ be a Banach space and $(x_n)$, $(y_n)$, $(f_n)$ be bounded sequences in $X$, $X$, $X^*$ respectively such that $f_m(x_n)=\delta_{mn}$ $\forall m,n$ and $\epsilon=\Sigma\|x_n-y_n\|<\infty$. ...
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### Isometric isomorphism

In the case that $L:B_1 \rightarrow B_2$ is a linear mapping of Banach spaces and $L$ is a isometric isomorphism (bijection and $\|Lx\|_{B_1} = \|x\|_{B_2}$) can I say that $L\overline{L}= 1$ is ...
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### A certain element which makes functionals positive

Suppose $X$ is a possibly non-separable Banach space and let $X^*$ be its dual. Also, let $(f_n)_{n=1}^\infty$ be a sequence of [EDIT: linearly independent] norm-one functionals in $X^*$. Does there ...
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### isomorphic properties

I need help with this proof: When $L$ is a isomorphic (bijection) linear mapping between two Banach spaces , in the case of both the spaces are Hilbert when using $L$, is it right to say that the ...
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### From weak and weak star to norm convergence

I haven't found this yet and I'm somehow not sure if my idea is correct. The Problem: Let $X$ be a separable Banach-Space, let $x_k\to x$ weakly and such that for every $\lambda_k \to \lambda$ weakly-...
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### Finite representability of $\ell_1$ and $c_0$

It follows from Krivine's theorem for a given Banach space $X$, either some $\ell_p$ or $c_0$ is finitely represented in $X$. Since finite dimensional $\ell_1$ and $\ell_\infty$-spaces are dual to ...
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### Sequences in Banach spaces [duplicate]

I am very bad with proofs that ask you to show that "$\exists$ something..." because in most of them you have to explicitly show the something. The following question is one such. Any help will be ...
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### Image of unit ball dense under continuous map between banach spaces

I am assuming that the following problem will require the open mapping theorem, or maybe the closed graph theorem. Any help that can be given will be deeply appreciated. The statement is the following:...
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### Can a proper vector subspace of a Banach space be a countable intersection of dense open subsets?

Let $V$ be a complete normed space. Let $W$ be a proper vector subspace. Can $W$ be the intersection of a sequence of open dense subsets of $V$? If there exists an open dense proper vector subspace ...
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### Hypervolume of a $N$-dimensional ball in $p$-norm
Suppose I have a N-dimensional ball with radius R in p-norm: $$\sum_{n=1}^N |x_n|^p = R^p$$ Is there a closed formula for its (hyper)volume? I can't find anything. If there isn't, can we at least ...
I was reading Reed and Simon Methods of Mathematical Physics Volume 1 and have a question about a small their proof of the open mapping theorem for Banach spaces. Let $T:X\rightarrow Y$ be a bounded ...