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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1answer
19 views

Relationships between one- and two-norms

I think this is quite a basic question but I'm struggling with it. Suppose $x_1, x_2, \ldots x_n$ are a set of finite-dimensional vectors and $a = (a_1, a_2, \ldots, a_n)$ and $b = (b_1, b_2, \ldots, ...
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Let $M$ be a finite dimensional subspace of $E$. Prove that for each $x \in E\setminus M$, there exists $m_0 \in E$ such that $d(x,M) = \| x-m_0\|$.

Let $M$ be a finite dimensional subspace of $E$. Prove that for each $x \in E\setminus M$, there exists $m_0 \in E$ such that $d(x,M) = \| x-m_0\|$. I've been stuck in these exercise for some time. I'...
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23 views

Properties of normed vector spaces which are Independent under translations

I am trying to prove the following proposition Let $X$ be a uniformly convex normed vector space and $A\subseteq X$ be a closed and convex subset, let $x \in X$. Show that there exists an unique $a ...
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45 views

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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51 views

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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1answer
22 views

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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56 views

Why isn't it obvious, from the definition of the essential supremum, that $\lvert f \rvert \leq \|f\|_{\infty}, \ \mu-$a.e?

Let $(X,\mathcal{A},\mu)$ be a measure space, and $f \colon X \to \mathbb{C}$ an $\mathcal{A}$-measurable function. My texbook defines, for essentially bounded functions with respect to $\mu$, i.e., $...
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The norm of a multilinear transformation between finite-dimensional vector spaces is always finite

I am studying n-linear maps from Zorich, Mathematical analysis II, p. 49-53, where the author writes that "it is not difficult to prove that for mappings of finite-dimensional spaces the norm of a ...
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Decomposition of a sequence of unit norm in $\ell^2$

For $1 \le p \le \infty$ let $S^p$ denote the unit sphere in $\ell^p(\mathbb{Z}^n)$, namely $x \in S^p \Leftrightarrow \| x\|_{\ell^p}=1$, and also set $B^p = \{ x \in \ell^p(\mathbb{Z}^n) : \| x\|_{\...
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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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31 views

Why the dual norm of a normed vector space is defined on the supremum on $||x||\leq 1$?

From a book I'm reading the definition of the dual norm of a n.v.s. $(X,||\cdot||)$ is defined as $||f||_{X^*}:=\sup_{||x||\leq 1}|f(x)|, x\in X$ where $f\in X^*$ So why is it not defined like $||f||...
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1answer
30 views

Total boundedness of $K \subset \ell^1$, iff $K$ is uniformly summable & bounded

Let $K \subset \ell^1$ and define uniform summability of $K$ such that for any $\varepsilon >0$ exists a natural number $N \in \mathbb{N}$ such that for every $x \in K$ the $N$-tail of the series ...
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1answer
33 views

Weighted matrix max-norm equivalent representations

Given the definition of weighted matrix max-norm as \begin{gather*} \left \| A \right \| _\infty ^w = \max \limits _{x \ne 0} \frac{\left \| A x\right \| _\infty ^w}{\left \| x\right \| _\infty ^w} ...
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How can I conclude that the weak topology on $~\mathcal l^2~$ is a proper subset of the norm topology from what I've done?

I'm doing a problem in topology. In a) I proved that the weak topology is coarser than the norm topology, and in b) I proved that the standard one sequence $~(e_n)~$ in $~\mathcal l^2~$ approaches $~(...
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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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37 views

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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Does equivalent norm imply the same order?

Consider two equivalent norms $|.|_{a}$ and $|.|_{b}$ on some vector space $V$. A general result then states that the norm induces the same topology on $V$. Does that imply that $$|x|_{a} \leq |y|_{a}...
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Proving $\| g^{-1}-f^{-1}\|\leq 2\|{f^{-1}}\|^2\|g-f\|$ [duplicate]

Let $f,g\in L(X,Y)$ where $X,Y$ are Banach spaces and let $f^{-1},g^{-1}\in L(Y,X)$. How to prove that $$\| g^{-1}-f^{-1}\|\leq 2\|{f^{-1}}\|^2\|g-f\|$$ if $\|g-f\|\leq\frac{1}{2\| f^{-1}\| }$?
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Inequality involving Unit Vectors

Let $x_1,...,x_n\in X$ a normed vector space and $\|x_i\|=1,\forall i\in\{1,...,n\}$. Suppose that for some $e\in(0,1)$ we have that $\|\sum_{i=1}^n\lambda_ix_i\|\leq (1+e)\max_{i\leq n}|\lambda_i|$ ...
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A finite dimensional topological vector space $V$ can be equipped with a norm $\|\cdot\|$ which gives the same topology.

Definition A topological vector space $V$ is a space equipped with a topology $\mathcal{T}$ for which the vector sum $+:V\times V\rightarrow V$ and scalar multiplication $*:\Bbb{K}\times V\...
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2answers
46 views

Showing incompleteness of metric space by specifying non-convergent Cauchy sequence

I have been looking at the space of continuous functions over a compact interval $C([0,2])$ equiped with the the integral norm of absolut values $\| \cdot \|_1$. I read a counterexample that showed ...
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1answer
34 views

Canonical projection on Quotient space maps open ball to open ball

I’m struggling to understand the last part of this proof where it says $\pi(x)=\pi(x-z)$ proves the claim. To prove the claim I suppose this must imply that $||x|| \leqslant ||x-z||$ so that $||x||<...
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31 views

Pre-Hilbert Spaces properties of norm

How can I prove that in a pre-Hilbert space $$||\lambda x + (1- \lambda) y|| = ||x||, \forall \lambda \in [0,1] \Rightarrow x = y$$. Also, is this property also true in normed spaces?
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Construct a vectors and dual vectors with given property

Let $X$ be an infinite-dimensional normed space (can assume Banach if needed). Given an integer $n\in\mathbb{N}$ and a mapping $f:\{1,\dots,n\}^2\to\{\pm 1\}$, I want to construct $\{x_1,\dots,x_n\}\...
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1answer
70 views

Unbounded linear operator between normed spaces

I am in the middle of a proof and this is one step I don’t understand Let $T:E\rightarrow F$ be a linear operator between normed spaces $E$ and $F$ If $T$ is unbounded then there exists a sequence $...
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1answer
51 views

Is my proof of the claim in example 5.1.7 in Notes on Elementary Linear Analysis (Bedos) correct?

I am trying to verify a claim in the example below, taken from these notes: https://www.uio.no/studier/emner/matnat/math/MAT4400/v19/pensumliste/ela-190523.pdf. Notation: $\ell^p(\mathbb{N}) = \{ (x(...
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57 views

Is the space of all bounded linear operators form a Banach space (under the given norm)?

Consider the class of all bounded linear operators on $X$ such that for each $f$ we have \begin{equation*} m(f)=\sup_{x\neq y}\frac{\left\vert f(x)-f(y)\right\vert }{\left\vert x-y\right\vert }. \end{...
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28 views

Show $C:=\{x\in V:\|x\|\leq 1\}$ is convex

Let $(V,\|\cdot\|)$ be a normed vector spcae and put $C:=\{x\in V:\|x\|\leq 1\}$. Show that $C$ is convex, which means that $0\leq t\leq 1$ $x,y\in C \Rightarrow tx+(1-t)y \in C$ My attempt: $\|tx+(...
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1answer
37 views

$L^p$ 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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Assuming inducing norm for given metric on a linear space

Let's start with a concrete example. Can I assume that for a metric $d(x,y) = \sqrt{|x-y|}$ on $\mathbb{R}^n$, there is only one possible norm to be assumed to induce this metric, say $\sqrt{|x|_2}= \...
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28 views

Equivalence of norms and a contradiction

I know the following theorem: Let $V$ be a real (or complex) finite dimensional vector space. Then all the norms on $V$ are equivalent. Take $V=\mathbb{R}^n$ on $\mathbb{R}$ and the metric $d_m(x,...
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Counter example for equivalent norms.

Let $V=\left \{ (x_n)_{n\in\mathbb{Z}}\in\mathbb{R}^\mathbb{Z}\mid sup_{n\in\mathbb{Z}}\left | x_n \right | <\infty \right \}$ I want to show that $\left\Vert \cdot\right\Vert ^{'}=sup_{n\in\...
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3answers
71 views

How to prove that whether it is a Banach space or not?

We consider the Banach space of all continuous functions on $X$ such that for each $f$ in the space, \begin{equation*} ||f||=\sup_{x\neq y}\frac{\left\vert f(x)-f(y)\right\vert }{\left\vert x-y\right\...
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Some related with $L^{2}$ norm and minimizing a function.

Let $X$ be a random variable which takes values on a $L^{2}$ functional space ($L^{2}(K)$ is the set of $L^{2}$ functions $f:K\rightarrow\mathbb{R}$, with $K$ be a compact set of $\mathbb{R}^{p}$). ...
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1answer
33 views

Closure of $C^\infty(\Omega)$ with respect to non-standard norm

Let $\Omega$ be an open bounded domain of $\mathbb R^d$ with smooth boundary $\Gamma$. I am wondering if it is possible to characterise the closure of $C^{\infty}(\overline\Omega)$ with respect to ...
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About the p-adic Cauchy sequences

I need to prove this: Show that if ($r_n$) is a $p$-Cauchy sequence of rationals and ($r_n$) is not converges to zero, then there exists $N\in$ N such that $|r_n|_p=|r_N|_p$ whenever $n>N$. ...
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Can a normed vector space be reconstructed from its metric?

Okay, me previous question Can a vector space be reconstructed form its norm? was solved by a nice (but in hindsight embarassingly simple) counter-example. Perhaps a better question emerges from that ...
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53 views

Can a vector space be reconstructed form its norm?

Let $(V,+,\cdot,\|.\|)$ be a normed vector space. Can we reconstruct addition $+$ of vectors and scalar multiplication $\cdot$ if we are given only the underlying set $V$ and the norm $\|\cdot\|\colon ...
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37 views

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?
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33 views

Example for a polynomially compact operator

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 ...
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1answer
21 views

$L^{2}$ inner product inequality.

Let $X$ be a random variable which takes values on $L^{2}$ space, $L^{2}(K)$ (the set of $L^{2}$ functions $f:K\rightarrow\mathbb{R}$ with $K$ a compact subset of $\mathbb{R}^{p}$). Let $f_{0}\in L^{2}...
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21 views

A complete metric space with some convex-type property

Let $(X,d)$ be a complete metric space with this property: for each $x \in X$, $r > 0$ and $y \in X$ with $d(x,y) < r$, there exists $z \in X$ such that $d(x,y)+d(y,z) = d(x,z) = r$. I want to ...
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1answer
25 views

Is this a sufficient definition for a linear functional on a subspace $Z$ of normed space $X$?

Let $X$ be a normed space over $\mathbb F$, and let $Y$ be a non-trivial subspace. Let $x_0 \in X$ be a non-zero element not in $Y$. Let $Z = \text{span}(x_0, Y)$, and define $f \in Z^\ast$ by the ...
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16 views

Infimum of square of $L^{2}$ norm and infimum of $L^{2}$ norm

Let $X$ be a random variable which takes values on a functional space, $L^{2}(I)$ (which is the set of $L^{2}$ functions $f: I\rightarrow\mathbb{R}$ with $I$ a compact set of $\mathbb{R}^{p}$). Let $...

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