# Questions tagged [normed-spaces]

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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### Understanding the Proof of the Hyperplane Separation Theorem

Am reading Wiki's proof of the Hyperplane Separation Theorem and am having trouble with the last part of the proof. Let me give you the structure of the argument and explain precisely where I have ...
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### find a non continuous complex homomorphism in non complete normed algebras

I want to show by an example there is a non continuous complex homomorphism in non complete normed algebras Can anyone give me an example? Thank you!!
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### Showing that $\|(x,y)\|_0=\sqrt{\|x\|^2+\|y\|^2}$ is norm if $\|\cdot\|$ is a norm.

The 3 properties are really easy to show but I cannot show that $\|(x,y)\|_0=\sqrt{\|x\|^2+\|y\|^2}$ satisfies the triangle inequality if $\|\cdot\|$ satisfies it. I tried to use Cauchy,S. inequality ...
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### Ficken's characterization of inner product spaces.

Ficken Characterization : Given X is normed space. For any vectors x,y in X and if $\Vert x \Vert = \Vert y\Vert$, then $\Vert ax+by\Vert = \Vert bx + ay \Vert$ , for any scalars a & b. The ...
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### Proof of $\lambda \in \sigma (T)$ implies $|\lambda|\leq ||T||_{B(X)}$

I want to show that $\lambda \in \sigma (T)$ implies $|\lambda|\leq ||T||_{B(X)}$ where $\sigma (T)$ is the spectrum of $T$ and $T\in B(X)$. I would like to check my proof here as it is different and ...
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### If closure of $C$ is a subvectorspace then so is $C$

Let $C$ be a convex subset of $\mathbb{R}^n$. It is asked to prove that if the closure $\overline{C}$ is a subvectorspace of $\mathbb{R}^n$ then so is $C$. Any ideas on where to start are welcome. ...
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### Operator norm inequality $\|XY\|\geq\frac{\|X\|}{\|Y^{-1}\|}$

Let $X, Y$ and $Y^{-1}$ be linear operators on a normed space. How to prove the inequality $$\|XY\|\geq\frac{\|X\|}{\|Y^{-1}\|}?$$ I already know that $\|XY\|\leq\|X\|\|Y\|$ but I don't see how I ...
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### Law of cosines in normed spaces which are not inner product spaces?

I have a finite dimensional normed vector space $V$ over $\mathbb{R}$. In practice I care mainly about the $p$-norm for $p\in[1,\infty]$, but there is no need to specialize to this case yet. I'm ...
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### Linear operator: existence of inverse, equivalence

We have just introduced linear operators in my FA class. Let $X,Y$ be normed spaces and $T:X \rightarrow Y$ a linear operator. Claim: $T$ has a continuous inverse $T^{-1}$ on $T(X)$, if and only if ...
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### Space of continuous functions form a Banach space?

Let $X$ be the collection of all continuous real-valued functions defined by \begin{equation*} ||f||=\sup_{x\neq y}\frac{\left\vert f(x)-f(y)\right\vert }{\left\vert x-y\right\vert } \end{equation*} ...
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### Is my proof of the separability of the closure of the range of a compact operator correct?

I want to prove the following remark (found in https://www.uio.no/studier/emner/matnat/math/MAT4400/v19/pensumliste/ela-190523.pdf) My attempt: Since $T$ is a compact operator, $T(B)$ is precompact ...
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### Can we find a point in the unit sphere that maps to a point who's projection onto the unit sphere is 'close' to the original point?

Given an non-zer0 $n\times{}n$ symmetric matrix $A$ and an $\epsilon$-net $\cal{N}$ of $S^{n-1}$. Is it possible to find a $x'\in\cal{N}$ s.t. $\bigg\|\frac{Ax'}{\|Ax'\|_2}-x'\bigg\|_2\le\epsilon$? I ...
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### $L^p$ for $0<p<1$ is not a normed space

I have been trying to prove that $L^p$ for $0<p<1$ is no normed space, however, researching quickly lead me to only find that these spaces are not locally convex. I am trying to understand this ...
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### The Relationship between norms and metrics

Are normed spaces a subset of metric spaces? As norms give rise to metrics is it right to say that the set of all normed spaces forms a subset of all metric spaces?
I understand to say that a bounded linear operator $T$ is called "polynomially compact" if there is a nonzero polynomial $p$ such that $p(T)$ is compact. Can anyone help me with examples of ...