# 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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### Is continuous extension on dense subset an isometry

If we have that $X \subset V$ is dense linear subspace. Where $V$ is normed space. I can show that for any $f \in X^{*}$, there exists a unique extension $\bar{f}$. I want to know if it can be shown ...
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### Where is the error in my proof?

I have this excercise. I am able to solve it, but the problem is that I can solve it without using the last part of information of the existence of the u-vector. That makes me afraid that my proof is ...
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### Existence of nontrivial bounded linear operator?

Are there two normed spaces such that there is no nontrivial bounded linear operator between them: $$\nexists T:X\to Y: T\text{ nontrivial, linear and bounded}$$
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### Equivalence of norms in finite-dimensional spaces

Assume we have a separable, reflexive Banach space $X$ such that $\{e_{i}\}$ is a basis. Let $X_{n} = \text{span}\{e_{1},\ldots,e_{n}\}$ be finite-dimensional subspaces where we define members ...
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### Let $V$be a vector space. Prove/Disprove: There is a norm $\|\cdot\|$, such that all subsets of $V$ are open sets in $(V,\|\cdot\|)$.

The Assignment: Let $V$ be a vector space over $\mathbb{R}$ with $V \not= \{0\}$. Prove or disprove: There is a norm $\|\cdot\|_d$ on $V$, such that all subsets of $V$ are open sets in ...
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### Natural example where $\ell_\infty$ distance appears.

The $\ell_2$ distance has a natural connotation: the straight line distance between two points "as the crow flies". Similarly, the $\ell_1$ distance has a natural connotation: the length of a path ...
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### Topology: Open, Closed Set and infinity norm

Since last week I've been learning a bit about Topology in Calculus and know the basic definitions of open, closed, norm, etc. Now I try to solve this question but I don't know how to. Its really ...
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### help about supremum norm proof

T:( $C'[0,1] ,||x||₁)\to (C[0,1]$,||x||∞) Tx(t)=x'(t) for any x is in C'[0,1] $||x||₁=||x||_{\infty}+||x′||$∞ how can we prove that equality? $||x||_1 = \sum_{n=1}^{\infty}|x_{n}|$ ...
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### Uniqueness of best approximation in strictly or uniformly convex normed linear spaces

I am studying approximation theory, just to learn basically, and in different books I saw a theorem about uniqueness of best approximation. One book uses strict, the other uses uniform convexity with ...
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### Bounding the distance between $L_\infty$ and $L_2$ for a continuous function

Consider a set of continuous (or even differentiable) functions $f_i(x)$, all defined for $x\in [a,b]$ for $i=1\ldots,N$. Can one define a uniform constant $c$ (which may depend on $f$) such that ...
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### Does completing a normed space commute with taking quotients?

Let $X$ be a normed vector space and $Y \subset X$ a closed subspace. We consider the quotient $X / Y$ and equip it with the quotient norm. Then we may form the completion $\overline{X / Y}$. We ...
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### Homework excercise, completeness in Vector-spaces, is it correct?, long, but can it be simplified?

I have a very difficult excercise. I see now that it became too much text for someone to might go through it, if you can please help me, but don't want to read all, can you please then only answer my ...
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### Why is one proof for Cauchy-Schwarz inequality easy, but directly it is hard?

Let's say you are in $\mathbb{R}^n$ and you define the norm as $||x||=\sqrt{x_1^2+x_2^2...+x_n^2}$. This we recognize as the usual norm from the inner product: $||x|| = \sqrt{\langle x, x \rangle}$, ...
### What norm on $\mathbb C (z)$
There are several different ways to define a norm on the space of polynomials $\mathbb C [z]$. For example, $\|p\| = \sup_{|z|\le 1}|p(z)|$ defines a norm. If $\mathbb C (z)$ denotes the field of ...