# 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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### counterexample for $C^1(U)$ not complete in any dimension

Cleary $C^1[a,b]$ is not complete with $\|\cdot\|_{\sup}$. I am looking for a counterexample which is working in any dimension, i.e. $C^1(U)$ is not complete for any open $U\subseteq \mathbb R^n$ ...
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### Interpolation of a subspace of codimension one

I am a little bit lost in interpolation theory. Let $A$ be a linear operator on $\mathbb{R}^{n}.$ Denote $V_0$ a subspace of $\mathbb{R}^n$ of codimension one. Suppose that $AV_0 \subseteq V_0$ and we ...
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### Weak closure of the boundary of the unit ball of $l^1$

I am trying to prove that zero is in the weak closure of set $\{ x\in l^1 \,:\, \lVert x\rVert =1\}$ . And I need example too. Is there any countable subset of this set which has $0$ in weak closure?
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### Convergence of series of linear operators in Banach space

Let $X$ be a Banach space and $A\in L(X,X)$. Show that $$\sum_{k=0}^\infty \frac{A^k}{k!}$$ converges in $L(X,X)$. Find an upper bound to the norm of sum. If a series converges absolutely, then it ...
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### Existence (and construction) of a convergent series

Suppose $(b_n)$ is an unbounded (real or complex) sequence. Does there always exist some (absolutely) convergent series $\sum a_n$ such that $\sum |a_nb_n|$ (or better, $\sum a_nb_n$) diverges? If so, ...
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### Not unique Hahn Banach extension

$G = \{(x_n) \in l_1: x_{2n+1} = 0, \forall n \in \mathbb{N} \}$ Let $f: G \to \mathbb{K}$ be a continuous linear functional, $f \neq 0$. Show that the Hahn Banach extension of $f$ is not unique. ...
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### Proof, that a set is not convex

I try to solve the problem 106 of the scottich book. I know the set of all rearranged sums is convex. Let $f_{j,k}$ the indicator function of the interval $(\frac{j}{2^k},\frac{j+1}{2^k}$). k = 0,1,2,...
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### Symmetric norms on $\mathbb{R}^n$ are between the $\ell^1$ and $\ell^\infty$ norms

We say that a norm $\|\cdot\|$ on $\mathbb{R}^n$ is symmetric if it is invariant under sign changes and permutations of the components. Suppose that $$\|(1, 0, ..., 0)\| = 1$$ for a symmetric norm. ...
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### $X$ be finite dimensional real NLS , let $x \in X$ , does there exist $T \in \mathcal B(X)$ such that $\{T^n(x):n \in \mathbb N\}$ is dense in $X$?

Let $X$ be a finite dimensional real normed linear space , let $x \in X$ , then does there exist a continuous linear transformation $T:X \to X$ such that $\{T^n(x):n \in \mathbb N\}$ is dense in $X$ ?
### $f \in \mathcal l^{\infty}{'}$ ; $f(x)\ge 0$ whenever $x \in \mathcal l^{\infty}$ is a sequence of non-negative terms ; is $f$ bounded? [duplicate]
Let $f:\mathcal l^{\infty} \to \mathbb R$ be a linear functional such that $f(x)\ge 0$ whenever $x \in \mathcal l^{\infty}$ is a sequence with non-negative terms ; then is $f$ continuous ?
### isomorphism between $C[0,1]$ and $C^1[0,1]$
Is space $C[0,1]$ with norm $\parallel f \parallel=\max|f(x)|$ (space of continuous functions on $[0,1]$) isomorphic to space $C^1[0,1]$ with norm $\parallel f \parallel=\max|f(x)|+\max|f'(x)|$ (space ...