# Tagged Questions

A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.

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### When is the completion of a space of functions a space of functions?

If $V$ is a $\mathbb C$-vector space of functions $f: X \to \mathbb C$ on some common domain $X$ and $\tau$ is a Hausdorff, locally convex topology on $V$, when may the completion of $(V,\tau)$ also ...
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### Preservation of the orthonormal operation in a linear transformation [closed]

Let $A: X\rightarrow X$ be a linear transformation, where $X$ is a inner product space. Show that, if $\|Ax\|=\|x\|$ for all $x\in X$, that $(x,y)=(Ax,Ay)$ for all $x,y \in X$. Show also that if ...
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### V is isomorphic to U. U is Banach iff V is Banach

$V$ is isomorphic to $U$ as normed vector space. $U$ is Banach if and only if $V$ is Banach. I don't know exactly, it seems easy at first look, but I have deep problem with the way I have to write the ...
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### What is the norm of matrices? Is it related to the norms of linear transformations? [closed]

What are the norms of a matrix? Is there any relation with norm of linear operators/transformations?
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### the space of lipschitz function is complete with respect to some norm

Let $V$ be the space of real valued lipschitz functions over $[a,b]$,we define: $M_f=sup_{x\neq y}\frac{|f(x)-f(y)|}{|x-y|}$ and lipschitz norm: $||f||_{Lip}=|f(a)|+M_f$ prove that $V$ with ...
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### Equivalency of two norms

Let $U$ be a normed vector space with two norms: $|| . ||,|| . ||^{'}$ For every sequence $\{x_n \}$ that $||x_n-x||\rightarrow 0$ & $||x_n-y||^{'}\rightarrow 0$,we can conclude $x=y$. ...
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### What values of $p$ give convergence to $0$ in $l^p$

Given a sequence $x_n \in l^p$ whose first $n^2$ members equal $\frac {1}{n}$, and all other entries $=0$, for what values of $p$ does the sequence converge to the zero sequence in $l^p$? So do I ...
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### Show which of the following norms are equivalent

On the vector space $C^1[0,1]$ of all real valued continuously differentiable functions defined in $[0,1]$, consider the following norms : $\displaystyle ||f||_{\infty}=\sup_{0\le x\le 1}f(x)$ , ...
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### What are we allowed to do when we know that two metric spaces are isometrically isomorphic?

What motivated the question Let $E$ and $F$ be two normed vector spaces and let $f: \Omega \subset E \to F$, where $\Omega$ is open. We denote by $\mathcal L(E,F)$ the set $\{L: E \to F, L$ is ...
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### Is there any standard name for this theorem about extension of bounded linear operators in normed spaces without changing the norm?

Let $X$ and $Y$ be normed spaces, both real or both complex; let, in addition, $Y$ be a Banach space; let $V$ be a (vector) subspace of $X$; let $T \colon V \to Y$ be a bounded linear operator; ...
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### In a normed vector space X: $x_n \to x$ weakly iff $d(x_n) \to d(x) ~\forall d \in D$, $D$ dense in $X^*$

Good day, I have the following task: Let $(x_n)_{n \in \mathbb{N}}$ be a sequence in the normed vector space $(X, || \cdot || )$ and let $x \in X$. Show that the following are equivalent: (i) $x_n$ ...
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### Open mapping theorem for normed abelian groups

A norm on an abelian group is a function valued in $\mathbb{R}_{\geq 0}$ which satisfies $|x|=0 \Leftrightarrow x=0$, $|{-}x|=|x|$, and $|x+y| \leq |x|+|y|$, not necessarily $|z x| = |z| |x|$ for ...
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### Exchanging limits with norms and linear functionals

In a normed vector space $X$, when can we say: $\lim\|x_n\|=\|\lim x_n\|$ and further, if $f\in X^{*}$, when can we say: $\lim fx_n=f(\lim x_n)$?
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### weak convergence in hilbert space and exchanging of limits

Question: Let $\{x_n\}$ be a sequence of elements of a Hilbert space $X$ which weakly converge to $x\in X$. Assume also that $\limsup\|x_n\|\leq\|x\|$ Show that $\|x_n-x\|\to0$. Proposed Solution: ...
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### Hahn-Banach and hyperplane separation

Let's take a look at the following result, based on Hahn-Banach (extension of bounded linear (real-valued) functionals): Let $X$ be a normed space, $U$ a subspace of $X$ and $u_0 \in X$ with ...
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### Normed Linear Space ,$p \neq 2$ is $\left \| f\right \|_{p}= \sqrt{f,f}$ for each $f \in L^P([0,1])$?

For $p \neq 2$, is there an inner product $< ., .>$ on $L^P([0,1])$ such that $\left \| f\right \|_{p}= \sqrt{f,f}$ for each $f \in L^P([0,1])$? is it true?for P=2 norm is induced by inner ...
### Operator norm: Show that $\|A\|=\sup\{ |f(Ax)| : x \in X, \|x\| \leq 1 , f \in Y^*, \|f\| \leq 1 \}$
Good day, As stated in the title, I have to show that $\|A\|=\sup\{ |f(Ax)| : x \in X, \|x\|_X \leq 1 , f \in Y^*, \|f\| \leq 1 \}$ where $\| \cdot \|$ is the operator norm, i.e. for $X,Y$ vector ...