Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.

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For what kind of matrix does it hold $\|XA\|_{1} \leq 1$ for a given $\|A\|_{1} \leq 1$.

All matrices are real. By $\| \cdot \|_1$ I mean a matrix norm induced by the vector norm $L_1$, i.e. the max of the column sums of absolute values. The matrix $A$ is given and we have $\|A\|_{1} ...
3
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1answer
25 views

Differentiation operator on smooth function with compact support

Suppose $f$ is $C^\infty$ with compact support. Let $T_n$ be the operator which sends $f$ to its $n$-th derivative. Is $||T_nf||_\infty$ bounded? It seems like I should use Stone-Weierstrass, but ...
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13 views

Norm of product of two matrices

Let $A\in\mathrm{R}^{n\times n}$ and $B\in\mathrm{R}^{n\times n}$ be two matrices. If $\|\cdot\|$ denotes the matrix norm, are the followings true? $\|AB\| = \|BA\|$ $\|A^2\| = \|A\|^2$ If they ...
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0answers
22 views

Show that $\{c \ge 0: ||Ax ||\le c ||x ||, \forall x \in X \} $ is closed , bounded from below and nonempty?

Let $A $ be a linear map from a normed linear space $X $ to a normed linear space $Y $ (both over the reals). How can I show that the set $\{c \ge 0: ||Ax ||\le c ||x ||, \forall x \in X \} $ is ...
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21 views

Should every element of a weighted norm vector be positive?

Suppose $\xi$ is a vector, that is used for $\parallel z\parallel_\xi$ calculation. Should every element of $\xi$ be positive, $\xi(i)>0$?
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23 views

Problem in proving norm

I have a linear space V that includes the continuous functions from [-$\pi, \pi$] to the complex set C, of the form: $$f(x) = a\cos(t)+b\sin(t) $$ where $a$ and $b$ are complex numbers. I want to ...
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1answer
45 views

Finding the norm of integral operator

I have the following operator: $A: (C^1[0;1];|||\cdot|||)\rightarrow(C^1[0;1];||\cdot||_{\infty})$ $Af(x)=\int_0^x f(t)dt$ where $|||f|||= ||f||_\infty+||Af||_\infty$ What is $||A||?$ I wrote ...
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0answers
19 views

Is there a single matrix norm such that for all stochastic $P$, $\| P \| = 1$?

By a stochastic matrix I mean a square real non-negative matrix with rows summing to one. Denote the set of all such matrices $\mathcal{S}$. By matrix norm I mean a norm in the vector space of ...
0
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1answer
34 views

Can a continuous linear form have a norm of infinity?

We know that a linear form $A \in V^{*}$ is continuous iff $$ \exists C: C\in R, ||Av|| \leq c ||v|| \forall v \in V $$ but we know too that $$ ||A|| = min\{c>0:||Av|| \leq C||v|| \forall v\in ...
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0answers
16 views

Color distribution distance

I am looking for a distance / squared distance between two color pixel distributions that would relate to the simple L2 distance when the color distribution is over one sample. The general problem I ...
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1answer
34 views

Any isometry is an isomorphism, though the converse is not true. [closed]

If we define a mapping $f:E \rightarrow F$, where $E$ and $F$ are normed vector spaces, then $f$ is an isometry if $f$ is a linear norm-preserving bijection, that is: $\|f(x)\|=\|x\|, \quad \forall x ...
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0answers
18 views

Proving that $\left(X(v,v)\right)^{\frac{1}{2}}$ defines a norm on a vector space where X is a positive-definite bilinear form.

Want to show: If $X$ is a positive-definite bilinear form on a vector space $G$ with real-valued scalars and $v\in G$, then $\left(X(v,v)\right)^{\frac{1}{2}}$ defines a norm on $G$. Thus far I have ...
2
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1answer
29 views

Norm on $\mathbb R^n$ with given unit ball

Consider a finite subset $S$ of $\mathbb Z^n$ such that $-s\in S$ whenever $s\in S$ and $S$ generates $\mathbb Z^n$. What is a norm on $\mathbb R^n$ whose unit ball is precisely the convex hull of ...
3
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0answers
51 views

Do these limits commute?

Given a sequence of functions $f_{n,m}:\mathbb{R}^{n} \to \mathbb{R}$, suppose that $$\displaystyle lim_{m} f_{n,m}(x)$$ exists almost everywhere (for any fixed n) and also suppose that ...
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1answer
40 views

Some Strange Minimum and proof

I read following on Norm chapter in one book. $$\begin{align}\left|\left\|x-y\right\|-\left\|w-z\right\|\right| \leq & \min \{\|x-w\| + \|y-z\| , \|x-z\| + \|y-w\|\}\\\text{ or, }&\min ...
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2answers
18 views

Norms — Distance Between Vectors

Which two of the vectors $u=(-2,2,1)^T$, $v=(1,4,1)^T$, and $w=(0,0,-1)^T$ are closets to each other in distance for (a) the Euclidean norm? (b) the infinity norm? (c) the 1 norm? I believe I know ...
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0answers
31 views

Is it possible to find the norm fuction of a space from an inner product already defined for it?

I'm a noob on the subject of functional analysis. As the title of the question says: Is it possible to find the norm fuction of a space from an inner product already defined for it? e.gr.: Suppose ...
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0answers
11 views

Equivalence of condition number from equivalence of vector norms

I must show that the equivalence of vector norms implies the equivalence of the condition number of its induced matrix norm. That is, given that for two arbitrary vector norms (+ and *) and an ...
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1answer
28 views

Why are the spectral norm of $A^{*}A$, $AA^{*}$ and $A$ equal?

I'm learning matrix norm now, but i don't have learned Hermitian before. Is there any theorem about hermitian i can use to prove that three matrices norm are equal?? Thanks a lot.
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26 views

Finding norm of functional coefficents in $C[a,b]$

Let $\{x_n\}$ denote Schauder’s basis for $C[a, b]$ and let $\{h_n\}$ be the associated sequence of coefficient functionals. Compute $||h_n||$. In young's book, a schauder basis for $C[a,b]$ ...
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1answer
31 views

Can you divide by matrix norms in an equation?

Supposing that a matrix A has an eigenvalue lambda, show that for any induced matrix norm, $||A|| \geq |\lambda|$. I attempted the solution, but I am not sure if it is valid to cancel the norm of ...
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30 views

A General Question arises about norms and limits [closed]

Suppose $x_n \to x_0$, i.e. $\lim x_n=x_0$. How we can show that $||x_n|| \to ||x_0||$, that is $\lim_{n \to \infty} ||x_n||=||x_0||$, so $\lim_{n \to \infty} ||x_n|| = ||\lim_{x \to \infty} x_n||$?
2
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1answer
32 views

$\|T^*T\|=\max\{\lambda: \lambda \text{ is an eigen value of } T^*T\}$

I want to prove this $\|T^*T\|=\max\{\lambda: \lambda \text{ is an eigen value of } T^*T\}$\ Suppose $\lambda_0>\lambda_1>\dots>\lambda_{n-1}\ge 0$ are distinct eigen values of $T^*T$ and ...
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2answers
9 views

Finding the norm of a complex trigonometric function?

Given that the complex norm $|z| = 1$, how would I go about proving that $|cos(z)| \leq e$? Just a hint would be helpful.
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0answers
51 views

Computing a “cheap” upper bound on the norm of the solution to a linear system

Consider the linear system $A x = b$, where $A$ is an invertible, $n \times n$, real matrix. I would like to compute a "cheap" upper bound on the (p-)norm of the solution; i.e. $\|x\|_p$. One can, of ...
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2answers
31 views

Projection function preserving distance

Given $n>1$. I wis to construct a function $f:\mathbb{R}^n\mapsto\mathbb R$ such that $$\|X_1-X_2\|\le\|Y_1-Y_2\|\implies|f(X_1)-f(X_2)|\le |f(Y_1)-f(Y_2)|$$ for all $X_1,X_2,Y_1,Y_2\in\mathbb{R},$ ...
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1answer
21 views

Cauchy Sequences Lemma in Vector Space E

I ran into a Lemma. Suppose $||.||_1$ and $||.||_2$ are two norms in vector spapce E, such that $||.||_1$ and $||.||_2$ are equivalent norms and {$x_n$} is an equivalent in E, then {$x_n$} is ...
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3answers
32 views

Cauchy Sequence some challenge

i read this sentence in one of math books: ‌Every convergent sequence in metric space is a cauchy sequence. would you please some one add more detail, why this is true? thanks.
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1answer
34 views

when norm of an operator is given by max of eigen values modulas

Could any one tell me how this $\|x\|^2=\|x*x\|$ and the rest of it? I know $\|x\|=\|x^*\|$, I also understand $x^*x$ is hermitian and so diagonalizale but then did not understand the norm square ...
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1answer
27 views

Norms for which every subset of closed unit ball containing the open unit ball is convex

It can be shown without much difficulty that any Euclidean norms satisfies the following condition :$$(P) \quad B \subset X \subset B' \Rightarrow X \, \text{is convex}$$ where $B=\{x \in E / \|x\| ...
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4answers
102 views

how to interpret this norm

Could any one tell me what "x" is there when he has defined $\|x\|$, just after he says $M_n$ has the operator norm thank you for helping.
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1answer
25 views

Strengthening bound in the Euclidean space

Suppose that $z$ is a $K\times 1$ vector. Denote the components of $z$ as $z_1,\ldots, z_K$. Let $r>0$ be given. I'd like to find the smallest constant $C$ such that $$ |z|^r\leq ...
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1answer
52 views

compactness of $L^2$ normed space

I have no idea how, where to start. I mean that we can show the compactness of the set via existence of convergent subsequence. But how can I take it? Please give a clue. This is my problem Show ...
4
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1answer
29 views

Prove some Equivalences Norm

Suppose $X=R^2$ and $x=(x_1, x_2)$. I see the following are equal EDIT: ( equivalence). why? i couldent find any proof to satisfy me. any hint or idea or proof highly appreciated. $||x||_1= |x_1| + ...
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1answer
22 views

Shortest distance from a point to a a Hyperplane

how could I prove the following using Lagrange optimization? Prove that the shortest distance from the hyperplane $$H= \{\vec{x} \in \mathbb{R}^{n} : \vec{a} \cdot\vec{x}=b\} $$ to a point ...
2
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1answer
46 views

Proof that the condition number of an isometry matrix is 1

At first glance it seems somewhat trivial, but I have some doubts, so I'd like your opinion. We are given that $\left\|Ax\right\| = \left\|x\right\|, ~ \forall x \in \mathbb{C}^{n}$ and want to show ...
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1answer
31 views

Norm of the Linear (Integral) Operator on $C[0,1]$

There may be some similar questions being asked but I didn't see anything that was quite what I was looking for so I'll ask the question here. Let $X = (C[0,1],||.||_{\max})$ and $T:X \to X = ...
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1answer
18 views

Help proof regarding sequence in subset of Hilbert space

I'm to prove the following: Let $H$ be a Hilbert space, and let $M$ be a non-empty convex subset of $H$. Suppose that $(x_n)$ is a sequence in $M$ such that $ ||x_n|| \to d$, where $d= \inf ...
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2answers
53 views

Is a minimum a minimum in every norm? (for finite-dimensional vector spaces)

I have a question regarding the equivalence of the norms in finite-dimensional vector spaces. Basically the question is: if $\hat{x}$ is some minimum-norm solution in a subspace $\mathcal{K}$ under ...
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1answer
31 views

Condition number - proof

I have a problem with which I've been struggling for a while. It's probably not that difficult, but I seem to be stuck so... here we are. Let A be an invertible lower- or upper-triangular matrix ...
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1answer
44 views

Norm on a Geometric Algebra

In the literature, for example "New Foundations for Classical Mechanics" by David Hestenes, the author introduces a function on the Geometric Algebra $$||M||^2=\langle M M^\dagger \rangle_0,$$ where ...
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1answer
22 views

Norm preserving Matrix properties

Norm-2 preserving can be done using unitary/orthogonal matrix: $A^*A = I => ||Ax|| = ||x||$ What is the matrix other than identity matrix that can preserve other norms ( norm-1, norm-inf) ?
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2answers
57 views

Norm equivalence (Frobenius and infinity)

I am trying to prove the matrix norm equivalence for norms 1, 2, $\infty$ and Frobenius. I have managed to solve find the constants for $||.||_{1}$ and $||.||_{2}$ but I cannot see how to continue ...
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2answers
49 views

Norm of the one dimension real space

I read that apart from a constant factor the absolute value is the only norm on the vector space $\mathbb{R}^1$. Why is that so?
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1answer
34 views

Lipschitz continuity and gradient of a real-valued function on a normed space

The gradient of a function $f:E\to\mathbb{R}$ is Lipschitz continuous with parameter $L > 0$ iff $$\|\nabla f(x) - \nabla f(y)\|^* \le L\|x-y\| \quad \forall x,y\in E.$$ I have two questions: ...
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1answer
36 views

Frobenius norm of a matrix [closed]

I know that Frobenius norm of a matrix A is equal to the square root of the trace of (A*conjugate transpose(A)). But how do I prove it mathematically?
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6 views

What will be the dual norm of $\Omega_p^\star := \max_{A \subset V , A \neq \phi} \frac{||S_A||_q}{F(A)^\frac{1}{q}}$

So, If $\frac{1}{p}+\frac{1}{q}=1$ and $F(A)$ is positive Real number for any $A \subset V$ and $F(A)=0$ for $A=\phi$. In that case what will be the dual norm of $\Omega_p^\star := \max_{A \subset V , ...
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1answer
39 views

Dual Optimization Problem

I have the following optimization problem, $$ \text{minimize}_{X,Y} \ \lVert X\rVert_* + \lambda \lVert Y\rVert_1 \\ \text{subject to }X + Y = C$$ $C \in \mathcal{R}^{m \times n}$, $\lVert ...
5
votes
2answers
122 views

Are there norms on $\Bbb{C}^m$ and $\Bbb{C}^n$ so that the norm $\Vert\cdot\Vert$ is a subordinate norm?

Denote $$\Vert A\Vert=\sum_{1\le j,k\le m}\vert A_{j,k}\vert$$ is cleary a norm over $M_{m,n}(\Bbb{C})$ but not a subordinate norm by taking the identity matrix $I$. So my question is: Can we make ...
0
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1answer
37 views

Is this a matrix operator norm?

Is the max element-wise norm a matrix operator norm? I know a matrix operator norm is defined by $$ |A|_p=\sup_{v≠0} \frac{|Av|_p}{|v|_p} $$ But how can I tell if the max norm is an operator norm?