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

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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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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
38 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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36 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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50 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
49 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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39 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
55 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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1answer
37 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
19 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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80 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
35 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
27 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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44 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
38 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
49 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
119 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
32 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
60 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 ...
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1answer
43 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
50 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 ...
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1answer
61 views

Find the norm of the operator $T:\ell^2 \to \ell^2$ defined by $Tx := (x_1, x_1+x_2, x_3, x_3+x_4, x_5, x_5+x_6, \ldots)$

Let $T : \ell^2 \to \ell^2$ (involving complex numbers) be defined by $$ Tx := (x_1, x_1+x_2, x_3, x_3+x_4, x_5, x_5+x_6, \ldots). $$ What is $\|T\|$? Essentially I've tried : To find $M \geq 0$ ...
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1answer
71 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
120 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
26 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
65 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
87 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
70 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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65 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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216 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
101 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
93 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
291 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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1answer
48 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 ...
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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 ...
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1answer
46 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?
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63 views

Finding the norm of a linear functional

This is a basic question of functional analysis, but I want to know how to... Find the norm of the linear functional $f$ defined on $C[-1,1]$ by $$f(x)=\int_{-1}^0 x(t) \, dt - \int_0^1 x(t) \, ...
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1answer
139 views

Inequality for subordinate norm of a submatrix : $\Vert B\Vert\le\Vert A\Vert$.

We 'did' this exercise in class Show that the subordinate norm of a submatrix is smaller than the the subordinate norm of the matrix i.e. $$\Vert B\Vert\le\Vert A\Vert$$ where $B$ is the sub ...
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2answers
33 views

Inequality $\vert x^{*}y\vert\le \Vert x\Vert_1\Vert y\Vert_{\infty}.$

I would like to prove that the following exercise : For all $x,y\in \Bbb{K}^n$ we have $$\vert x^{*}y\vert\le \Vert x\Vert_1\Vert y\Vert_{\infty}.$$ Where $\Bbb{K}=\Bbb{R}$ or $\Bbb{C}$ and ...
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1answer
57 views

Subordinate norm is equal to $\Vert A\Vert_M=\Vert MAM^{-1}\Vert$ for the norm $\Vert x\Vert_M=\Vert Mx\Vert$

Let $M\in \mathcal{M}_n(\Bbb{K})$ a nonsingular matrix, we have $\Vert x\Vert_M=\Vert Mx\Vert$ is a norm over $\Bbb{K}^n$. I have to prove that the subordinate norm is equal to $\Vert A\Vert_M=\Vert ...
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Proof of Matrix Norm (Inverse Matrix)

Show for any induced matrix norm and nonsingular matrix A that $$ \left\|A^{-1}\right\| ≥ (\left\|A\right\|)^{-1} $$ where $$ \left\|A^{-1}\right\| = ...
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1answer
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Show that $\infty$-norm and $C^1$-norm are not equivalent.

Show that $\infty$-norm and $C^1$-norm are not equivalent. For the $C^1([a,b],\mathbb{R})$ space, show that $\displaystyle ||g||_\infty=sup_{a\leq t\leq b}|g(t)|$ and $\displaystyle ...
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1answer
22 views

Matrix norm square properties.

I'm trying to prove one of these inequalities. This isn't a homework problem but trying to solve out of curiosity as it didn't have any relationship between $x$ and $\alpha$. How do you prove: ...
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1answer
25 views

Norm2 of a vector of complex numbers

I am migrating a matlab code into C++ and I need to know how does matlab calculate the norm of below matrix. For two numbers, A=a+ib , B=c+id, I know I should do [(a-c)^2+(b-d)^2]^1/2. But how is it ...
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3answers
137 views

Is it true that $\biggl\|I-\frac{vv^T}{v^Tv}\biggr\|=1$?

I am following a proof in the text OPTIMIZATION THEORY AND METHODS a springer series by WENYU SUN and YA-XIANG YUAN. I come across what seems obvious that for a column vector $v$, with dimension ...
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1answer
37 views

How to prove that the L2 norm is a non-increasing function of time for a 2nd-order PDE?

I am having a test in few days and I saw an interesting question while I was skimming through the book problems. The problem is concerned about initial-boundary value problem of 2nd order PDEs. To ...
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1answer
55 views

Operator norm and Hilbert Schmidt norm

I'm looking for a proof of \begin{equation} ||T||\leq ||T||_{HS}, \end{equation} for which it is sufficient to show \begin{equation} ||Tx|| \leq ||x|| \cdot ||T||_{HS} \forall x\in H, x\not=0 ...
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1answer
47 views

An inequality using Sobolev norms

Let $\| \cdot \|_{H^s(\mathbb R)}$ be the usual Sobolev norm in $\mathbb R$ and $r>0$. If we have $$ \|f\|_{L^\infty(\mathbb R)} \le \| f\|_{H^k(\mathbb R)} $$ for all $k>r$, then the inequality ...
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1answer
26 views

Distance of two vectors under L_inf norm

Very simple question: suppose I have two vectors $a = (1,-2)$ and $b = (4,2)$. Under the L_inf norm, would the distance between them be $abs( ||a||_{inf} - ||b||_{inf}) = abs(2 - 4) = 2$? Is this the ...
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Norm of an integral operator

I have an exercise that I need to solve and I can't finish it. Let $k \in \mathcal{C}([0,1] \to \mathbb{R})$. Proove that this operator : $$ \begin{array}{ccccc} T & : & ...