# 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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### If $T:X \rightarrow Y$ is a linear operator and $r>0$ such that $r \cdot B_Y \subseteq T(B_X)$, show $y ||x|| \leq M ||y||$.

Let $X$ and $Y$ be normed spaces and let $B_X$ and $B_Y$ denote the closed unit balls in $X$ and $Y$ respectively. Suppose $T:X \rightarrow Y$ is a linear operator and that there is an $r>0$ such ...
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### Counterexample for the stronger statement of Riesz's lemma

Here is a counterexample for the stronger statement of Riesz's lemma and I don't understand it. Why for all $x$, such that $||x||=1$, there exists $y \in Y$, such that $d(x,y)<1$?
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### Are these conditions also sufficient for a metric to be induced by a norm?

Let $(X,d)$ be a metric space such that the set $X$ is also a vector space over the field $\mathbb{R}$ of real numbers or the field $\mathbb{C}$ of complex numbers. Then the following holds: If ...
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### Isometries in metric $L_n$

Let's define distance between the points in 2-d space: $$d = \sqrt[n]{(x_1 - x_2) ^ n + (y_1 - y_2) ^ n},\quad n > 2$$ The isometries (linear transformations that preserves the length of vectors) ...
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### Closure of a subset of normed vector space

Can you help me to prove this claim : $A$ is a subset of a normed vector space, closure of $A$ is closure of $$B=\bigcap_{n=1}^\infty \left( A+{1\over n}B_1 (0)\right)$$ I tried to prove ...
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### $\| f \|_{BV} = | f(a) | + V_{a}^{b} f$ defines a norm in the space $BV[a,b]$

I'm learning about functions of bounded variations and need to verify my work for this problem: Show that $\| f \|_{BV} = | f(a) | + V_{a}^{b} f$ defines a norm in the space $BV[a,b]$. My ...
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### Let $E$ be a normed space and $b\in E$. Then $d(b, \overline{X}) = d(b,X)$.

$d$ is the distance between a point and a set: $d(b,X) = \underset{x\in X}{\inf}\{\|b-x\| \}$ and $X = B(a;r) = \{\|a-x\|<r: x \in E\}$, $\overline{X} = B[a;r] = \{\|a-x\| \leq r: x \in E\}$ ...
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### boundedness of convex functions

Let $X$ be a vector space, $\Omega$ a convex subset thereof and $f:\Omega \to \mathbb R$ a convex function. Then $f$ need not be bounded from below - not even if it is strictly convex, as the example ...
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### Can derivative of a smooth norm be zero?

Let us call a norm on $\mathbb{R}^n$ smooth if its restriction $\| \cdot \|:\mathbb{R}^n\setminus \{ 0 \} \to \mathbb{R}$ is a smooth map. Is it true that its differential at (every non-zero point) ...
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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 $A$...
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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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### 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; and ...
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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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### The set $A=\{ x :\|x\|=1 \}\subset \mathbb{R}^n$ is compact in $\mathbb{R}^n$ under euclidean norm.
The set $A=\{ x \mid \|x\|=1 \}\subset \mathbb{R}^n$ is compact in $\mathbb{R}^n$ under Euclidean norm. If I can show this is closed and bounded then we are done. It seems bounded but I am not sure ...
### Let $A=\{ f \in \ell^1\, :\,\text{ for each natural number}\, n\,\text{we have}\, |f(n)|<1/2^n \}$. find the closure of $A$ in $\ell^1$.
Let $A=\{ f \in \ell^1\, :\,\text{ for each natural number}\, n\,\text{we have}\, |f(n)|<1/2^n \}$. find the closure of $A$ in $\ell^1$. I know the interior of A is empty. for the case of getting ...