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

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Linear isometry between $c_0$ and $c$

The following question is an exercise and so I'm just looking for advices and not for answers if it's possible. I have the following sets in $l^\infty$ $$c_0 := \{x_n \in l^\infty: \lim x_n = 0\} ...
4
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1answer
247 views

Proof of an inequality of $L^p$ norms

For a general measure space, we define : $\|f\|_p= \left(\int\vert f\vert^p du\right)^{1/p}$. Let $0 < a < b < c < \infty$ and prove the following: $$ \|f\|_b \leqslant \max\{\|f\|_a, ...
4
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1answer
213 views

Finding the norm of the operators

How do I find the norm of the following operator i.e. how to find $\lVert T_z\rVert$ and $\lVert l\rVert$? 1) Let $z\in \ell^\infty$ and $T_z\colon \ell^p\to\ell^p$ with $$(T_zx)(n)=z(n)\cdot ...
4
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335 views

Proving two results about the spectral radius

How do I prove these two theorems? Furthermore, can I apply them to infinite-dimensional spaces, such as Banach spaces? Theorem 1. Let $M\in \mathbb{C}_{n\times n}$ be a matrix and $\epsilon ...
4
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1answer
285 views

$\|f+g\|_p\leq\|f\|_p+\|g\|_p$ where $\|f\|_p=(\int_{a}^{b}|f(t)|^p dt)^\frac{1}{p}$

Using the fact that $$cd\leq\frac{c^p}{p}+\frac{d^q}{q}$$ if $$\frac{1}{p}+\frac{1}{q}=1$$ and letting $$c=\frac{|f(t)|}{\left(\int_{a}^{b}|f(t)|^p dt\right)^\frac{1}{p}}$$ and ...
4
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2answers
46 views

Minimizing the Frobenius Norm

I would like to minimize the following expression with respect to matrix X: $$\left \| A-BX \right \|_{F}$$ where A and B matrices are given and all the matrices have positive integer elements. Any ...
4
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1answer
158 views

Reproducing Kernel Hilbert Space (RKHS) constructed by the summation of positive-definite kernels.

Let $k_1,\ldots,k_p$ be positive definite kernels defined on $\mathcal{X}\times\mathcal{X}$, where $\mathcal{X}$ is a non-empty set. $k_i$ is the reproducing kernel of the Reproducing Kernel Hilbert ...
4
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3answers
579 views

Monotonicity of $\ell_p$ norm

Consider a $n$ dimensional space, it is known (Wikipedia) that for $p>r>0$, we have $$ \|x\|_p\leq\|x\|_r\leq n^{(1/r-1/p)}\|x\|_p. $$ I have two questions about the above inequality. $(\bf ...
4
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2answers
858 views

I am not sure how to calculate this norm?

I have the following matrix: $$A= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{bmatrix} $$ What is the norm of $A$? I ...
4
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1answer
232 views

$L_1$ projection of sum of convex functions onto polytopes

Suppose I have a function $f(x) : \mathbb R^n \to \mathbb R$ that is the sum of a given strictly convex function $g : \mathbb R \to \mathbb R$ in a single variable, i.e. $f(x) = g(x_1) + g(x_2) + ...
4
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1answer
633 views

What are norms of sub-matrices invariant under a block diagonal similarity transformation of a block matrix?

Say $M := \begin{pmatrix} A & B\\ C & D \end{pmatrix}$ is a block matrix with $A, D$ being square matrices and this $B$ and $C^T$ having the same shape. Is there any norm characterizing the ...
4
votes
1answer
43 views

Is this sufficient for continuity?

Assume you have a map $\phi : V \rightarrow \mathbb{C}$, where $V$ is a complex vector space. Now, if we have $\phi(\lambda x) = |\lambda | \phi(x)$ and $\phi(x+y)^2+ \phi(x-y)^2 = 2\phi(x)^2 +2 ...
4
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31 views

Estimate the Cauchy integral for matrix-valued functions

Recently, I became familiar with the concept of the "matrix function via Cauchy integral", i.e., $$f(A):=\frac{1}{2\pi i}\int_\varGamma f(z)(zI-A)^{-1} \mathrm{d}z$$ Furthermore, it can be shown that ...
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2answers
87 views

let $A$ be an n by n matrix, show that $||A||_{OP} \leq ||A||_{HS} \leq \sqrt{n} ||A||_{OP}$

We are given $A \in M_{n}(\mathbb R)$ and the following norms: $||.||_{e}$ is the standard euclidean norm of $\mathbb R^n$. $||A||_{OP}$ is the operator norm of $A$, meaning $||A||_{OP} = ...
4
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1answer
173 views

Generalization of $\frac{a + b}{c + d} \leq \text{max}(\frac{a}{c}, \frac{b}{d})$

I'm looking for a matrix version of the basic inequality for the ratio of two sums of positive numbers: $$\frac{a + b}{c + d} \leq \max\left\{\frac{a}{c}, \frac{b}{d}\right\}.$$ Specifically, I have ...
4
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231 views

How to determine a operator norm?

How to solve following: In Hilbert space $W_2^1=\{f:[0,1]\rightarrow \mathbb{C}|f\in AC[0,1], f'\in L^2[0,1]\}$ with scalar product $(f,g)=\int_0^1 f\overline{g}dx+\int_0^1 f'\overline{g'}dx$ is ...
4
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2answers
71 views

norm of inverse less than 1

I just wanna ask if what I am doing here make sense: Let $\epsilon$ be arbitrary positive number. Choosing $\epsilon$ and let it approaches 0, I would like to have $||(I-\epsilon A)^{-1}|| < 1$. ...
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1answer
205 views

Matrix Norm set #2

As a complement of the question Matrix Norm set and in order to complete the Problem 1.4-5 from the book: Numerical Linear Algebra and Optimisaton by Ciarlet. I have this additional conditions: (3) ...
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402 views

Poincaré's lemma with norm in $H_{0}^{1}$

I wonder why $\| u\|_{H_0^1} = \int_{\Omega} |Du|^2$ for u in $H_0^1(\Omega)$, with $\Omega = (-1,1)$? I might be wrong but isn't $\| u\|_{H^1} = \| u\|_{L^2} + \| Du\|_{L^2}$? How come that $\| ...
4
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2answers
358 views

$L^p$ norm of multivariate standard normal random variable

Given $X_i\sim \mathcal{N}(0,1)$ what is the behaviour of $$ ||X||_{l^p}=(\sum_{i=1}^n|X_i|^p )^{1/p}$$ as $n\rightarrow \infty$? For $p=2$ results about $\chi$-distribution tell us that ...
4
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1answer
338 views

Prove a basic fact on a linear combination of vectors

Let $x_i (i=1,...,n, n>d)$ be a unit vector in $R^d$. $c_i>0$ is a positive real scalar. How to prove the following fact? Fact: There exist some vectors $x_i$ such that $\sum_{i=1}^n c_i ...
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1answer
64 views

(still open) For what $X$ do we have $\|XA\|_{1} \leq 1$ 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} ...
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59 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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53 views

Find a relationship between $\hat f$ and $\hat g$, given $g(x)=\int^{x+1}_x f(t)\,dt$, $f\in L^2(\mathbb R)$,

Given $f\in L^2(\mathbb R)$ and $g(x)=\int^{x+1}_x f(t)\,dt$ (a) Relate $\hat f$ and $\hat g$ (b) Prove $g\in H^1(\mathbb R)$ Part A: $$\hat g(k)= \int^{\infty}_{-\infty} \int^{x+1}_{x} ...
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124 views

norm of differential operator on $P^n[0,1]$

Consider the space $P^n[0,1]$ of polynomials of degree $\leq n$ on $[0,1]$, equipped with the sup norm. Now, this is a finite dimensional space, so all linear operators have to be continuous, hence ...
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179 views

Effect of doubly stochastic matrix on vector norm

Let $D$ be a $N \times N$ doubly stochastic matrix, $x$ be a $N$ dimensional vector. What is the relation between $\Vert Dx \Vert_2$ and $\Vert x \Vert_2$? In addition if $\Vert x \Vert_2=1$, what ...
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0answers
117 views

What are norms used for?

These two questions are quite similar to this one, so I apologise if this irritates anyone. Also, I suspect that a lot can be said in the answer, so I am really just looking for some main points ...
4
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3answers
539 views

Example of a sequence that converges to two different limits with respect to two complete norms

I've wondered about the following question : Is there an (explicit?) example of a vector space $X$, two complete norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on $X$, and a sequence $(x_n) \subseteq X$ such ...
3
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Defintion of $L_\infty$ norm

Where does the definition of the $L_\infty$ norm come from? $$\|x\|_\infty=\max \{|x_1|,\dots,|x_k|\}$$
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569 views

Normalizing a matrix

I came across a step in an numerical algebra algorithm that says "Normalize the rows of matrix A such that they are unit-norm. Call U the normalized matrix." I do something like this: ...
3
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1answer
3k views

The $ l^{\infty} $-norm is equal to the limit of the $ l^{p} $-norms. [duplicate]

If we are in a sequence space, then the $ l^{p} $-norm of the sequence $ \mathbf{x} = (x_{i})_{i \in \mathbb{N}} $ is $ \displaystyle \left( \sum_{i=1}^{\infty} |x_{i}|^{p} \right)^{1/p} $. The $ ...
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938 views

equivalent norms in Banach spaces of infinite dimension

Suppose $ X $ is a Banach space with respect to two different norms, $ \|\cdot\|_1 \mathrm{ e } \|\cdot\|_2 $. Suppose there is a constant $ K > 0 $ such that $$ \forall x \in X, \|x\|_1 \leq ...
3
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63 views

whats the difference between $|v|$ and $||v||$?

$v$ being a vector. I never understood what they mean and haven't found online resources. Just a quick question. Thought it was absolute and magnitude respectively when regarding vectors. need ...
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3answers
1k views

Example of two norms on same space, non-equivalent, with one dominating the other

I am looking for an example (with proof) of two norms defined on the same vector space, such that the norms on the two spaces are NOT equivalent, but such that one norm dominates the other...
3
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1answer
87 views

Does Permuting the Rows of a Matrix $A$ Change the Absolute Row Sum of $A^{-1}$?

For $A = (a_{ij})$ an $n \times n$ matrix, the absolute row sum of $A$ is $$ \|A\|_{\infty} = \max_{1 \leq i \leq n} \sum_{j=1}^{n} |a_{ij}|. $$ Let $A$ be a given $n \times n$ matrix and let $A_0$ ...
3
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1answer
373 views

Frobenius norm is not induced

My textbook says: Frobenius norm defined on $\mathbb{R}^{m,n}$ by a formula: $$\|A\|_F=\sqrt{\sum_{i=1}^n\sum_{j=1}^m|a_{i,j}|^2}$$ when $n>1, \ m>1$ is not induced by any vector's ...
3
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2answers
83 views

Norm of Hilbert matrix is it equal to $\pi$?

Let $A$ be a Hilbert matrix, $$a_{ij}=\frac{1}{1+i+j}$$ We have the following result : $\Vert A\Vert\leq \pi$. I am using the subordinate norm of the euclidean norm i.e. $$ \Vert A\Vert=\sup\{\langle ...
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1answer
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Why is the operator norm of a diagonal matrix it's largest value?

I read this in my textbook have tried working through it - I keep getting max 2-norm(Ax), which is just the magnitude of Ax. How should I do this proof? (note, this is not for homework, I'm just ...
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124 views

Is $l^2$ norm differentiable at $x=0$?

For $l^2$ norm on $\mathbb{R}^d$, $\frac{d\|x\|_2}{dx} = \frac{x}{\|x\|_2} $, so $d\|x\|_2$ is differentiable wherever $x \neq 0$. is the norm differentiable at $x=0 \in \mathbb R^d$? Thanks!
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315 views

Norm of differentiation operator $Tf(t)=f^{'}$..

Consider $T:C^1[0, 1]\rightarrow C[0, 1]$ given by $Tf=f'$ where $$\|f\|_{C^1}=\|f\|_\infty+\|f'\|_\infty$$ and $\|f\|_\infty=\sup_{x\in [0, 1]}|f(x)|$. How to prove $\|T\|=1$? The inequality ...
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132 views

Linear Algebra munkres analysis on manifolds question.

If $A$ is an $n$ by $m$ matrix and $B$ is an $m$ by $p$ matrix, then $$ |AB| \leq m|A||B|$$ where $|A| = \max\{|a_{ij}| : i = 1,\ldots,n \text{ and} j = 1,\ldots,m\}$ Attempt: $ |AB| = \max\{| ...
3
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174 views

Inequality involving norm and inner product

I am stuck proving this trivial inequality: on a real inner product space, $(||x||+||y||)\frac{\langle x,y\rangle}{||x|| \cdot ||y||}\leq||x+y||$ I have tried to square both sides and use the Cauchy ...
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229 views

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\| = ...
3
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182 views

Poincare Inequality implies Equivalent Norms

I am currently working through the subject of Sobolev Spaces using the book 'Partial Differential Equations' by Lawrence Evans. After the result proving the Poincare Inequality it says the following ...
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97 views

When does $\|x+y\|=\|x\|+\|y\|?$

Let $(V_\mathbb R,\langle,\rangle)$ be an inner product space. I'm trying to see for $x,y\in V$ when does $\|x+y\|=\|x\|+\|y\|?$ Let $\|x+y\|=\|x\|+\|y\|$ Squaring both sides, $\langle ...
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120 views

Geometric Sums in Banach Algebra

Let $E$ be a Banach Algebra with identity, and $v\in E$, so that $||v|| < 1$. The geometric series $w = \sum_{k=0}^\infty v^k$ converges in the norm. I can show that $||w|| \le ...
3
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2answers
175 views

How is $L_{2}$ Minkowski norm different from $L^{2}$ norm?

I am reading the book Multidimensional Particle Swarm Optimization for Machine Learning and Pattern Recognition. They use $L_{2}$ Minkowski norm (Euclidean) as the distance metric in the feature space ...
3
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2answers
75 views

Upper bound for norm of Hilbert space operator

It is a standard result that for a bounded self-adjoint operator $T$ on a complex Hilbert space $H$, we have $||T||=\sup_{||x||=1}|\langle Tx,x\rangle|:=M$. It seems that for any bounded operator on ...
3
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2answers
61 views

Norms of eigenvalues bigger than 1 implies $|Ax|>x$ for all nonzero $x$?

If all the eigenvalues of $A$ (an n by n real matrix) have norms bigger than 1, is it true that $|Ax|>|x|$ for all nonzero $x\in\mathbb{R}^n$? (This is clearly true if $x$ is an eigenvector ...
3
votes
1answer
69 views

Showing that the zero vector has norm zero

I need to show that this is a property of a norm. I know this is supposed to be straightforward but I am somehow not seeing it. The property is $$\lVert 0\rVert = 0$$ I was trying to use the fact ...