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

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help about supremum norm proof

T:( $C'[0,1] ,||x||₁)\to (C[0,1]$,||x||∞) Tx(t)=x'(t) for any x is in C'[0,1] $||x||₁=||x||_{\infty}+||x′||$∞ how can we prove that equality? $||x||_1 = \sum_{n=1}^{\infty}|x_{n}|$ ...
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36 views

Hilbert Schmidt norm inequality

I was wondering if anyone knows about an inequality for the Hilbert-Schmidt (H-S) norm of the type $|Tr (Bg)|\leq Const\cdot||B||\cdot function(||g||_{2})$ for a bounded operator $B$ and a H-S ...
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47 views

Inequality of Weighted norm

I have a question about the weighted norm inequality: The weighted norm of a vector $x\in R^{M\times N}$ is defined by: $\left \| X\right \|_{w,*} = \sum_{_{i}}\left |w_{i}\sigma _{i}\left ( X ...
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34 views

Questions over a specific case of the Muntz-Szasz theorem proof

On page 157 of this site: http://arxiv.org/pdf/0710.3570.pdf the author is proving a specific case of one direction of the Muntz-Szasz theorem. I do not understand the following 3 claims: 1) For ...
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Showing 2- norm of Ax,squared is equal to $$ x^TM^TMx $$

So, I am trying to prove $$\|Mx\|^2 =x^TM^TMx$$, however I am running into some difficulties. Here, $M \in \mathbb{R}^{m \times n}$ and $x \in \mathbb{R}^n$. I know that when you take the transpose ...
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47 views

Norm of an operator defined on sequence spaces

Consider the sequence space $\ell_r$ defined by $$\ell_r=\left\{x=(x_n)_{n=1}^{\infty}:x_n\in\mathbb{R}\text{ and }\sum_{n=1}^{\infty}|x_n|^r<\infty\right\}.$$ Let $2\leq p,q<\infty$ such that ...
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22 views

Bounding the distance between $L_\infty$ and $L_2$ for a continuous function

Consider a set of continuous (or even differentiable) functions $f_i(x)$, all defined for $x\in [a,b]$ for $i=1\ldots,N$. Can one define a uniform constant $c$ (which may depend on $f$) such that ...
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42 views

limit of p norm as p goes to 0!

Suppose we have a measure $\mu$ and a space $X$ such that $\mu(X)=1$, and a function $f \in L^r$ for some $r > 0$, where $L^r$ is defined in the usual way even for numbers less than $1$. Show ...
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1answer
54 views

Are the following norms equivalent?

We have the norms $||f||_1=||f||_\infty+||f'||_\infty$ and $||f||_2=|f(a)|+||f'||_\infty$ where $f\in C^1[a,b]$. Are they equivalent and how shoud I prove/disprove this.
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43 views

Question on the norms

I got stuck with the following (simple) question since the result I got seems to be counterintuitive: I have a function defined in terms of its Chebyshev expansion, i.e. ...
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17 views

A simple inequality with two sequences.

Suppose there are two sequences $a^n\in \mathbb{R}^p,b^n\in \mathbb{R}^d$, $n=1,2,\cdots,+\infty$. $K\in \mathbb{R}^{d\times p}$ is a matrix and let $C<\|K\|$. The following inequality always ...
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31 views

Ratio of $\|\cdot\|$ and $\|\cdot\|_{\infty}$ on $\mathbb{R}^2$

I have the following question from an old examination paper in Real Analysis: On $\mathbb{R}^2$, $\|\cdot\|_{\infty}$ is defined by ...
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35 views

an $L^p$ implication

Let $1<p<\infty$ and $x,y\in L^p([0,1])$ such that $\|x\|_p = \|y\|_p = 1$. Then the following implication holds: $$\left\|\frac{x+y}{2}\right\|_p=1\Rightarrow x=y\tag{*}$$ This ...
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42 views

Frechet derivative of the sup norm function on $C[0,1]$

I know that the derivative in the title does not exist for any x but I do not have a clue why. Could someone explain why this derivative DNE at 0 and I can figure out why it does not exist at x? ...
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1answer
56 views

Estimate for weak $L^{1}$ norm

Let the weak $L^{1}$ norm on $f$ be defined by $\|f\|_{\mathrm{WL}^{1}} = \sup_{t > 0}t D_{f}(t)$ where $D_{f}(t) = \mu(\{x \in \mathbb{R}: |f(x)| > t\})$ and $\mu$ is the standard Lebesgue ...
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34 views

Is this some kind of triangle inquality?

I stumbled upon the following inequality: $$\Vert x+hz-(x+y)-(p-(x+y))\Vert_2 \geq \Vert p-(x+y)\Vert_2-\Vert x+hz-(x+y)\Vert_2$$ where $p,x,y,z \in \mathbb{R}^n$. My question is: Is this some kind ...
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28 views

norm on a quotient-space

Let $M:[0,\infty)\to[0,\infty)$ be continuous and convex. Further $M$ satisfies $M(t)=0\Leftrightarrow t=0$. Let $$\mathcal L_M(\mathbb R):=\left\{f:\mathbb R\to\mathbb R \mathrm{\ measurable\ ...
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22 views

Spectral Norm of $2\times 2$ symmetric matrix

Consider a $2\times 2$ symmetric matrix, in this case, is there some closed formula for its spectral norm ? By spectral norm I mean the induced 2-norm, there is a definition here. Thanks.
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43 views

lp distance, p tends to minus infinity

Minkowki distance as p tends to minus infinity equals to the smallest difference along any coordinate dimension. Is this difference a metrics? Even if the lp norm with p tending to minus infinity is ...
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24 views

Metric induced from norm

I was trying to understand the following: Every norm on $R^n$ is continuous (as a map from $R^n$ to $R$). Proof. We use the maximum metric on $R^n$: $ d(x, y) = \max{|x_j − y_j| : j ∈ \{1, . . . ...
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26 views

Check continuity of linear functionals and find norms

1) $c_{00} \owns (x_n) \mapsto \sum_{n=0}^{\infty} x_n \in \mathbb{K}$ where $c_{00}$ is a space of sequences that are eventually equal to $0$ with sup norm 2) $\ell^\infty \owns (x_n) \mapsto ...
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23 views

Norm and Matrice Proof

I'm trying to show that the following statement is true: If $||\mathbf a - \theta \mathbf b||^2 - ||\mathbf a||^2 \geq 0$ for all $\theta \in [0,1]$, then $\mathbf a^T \mathbf b \le 0$. Is this ...
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61 views

How to prove that the spectral radius of a linear operator is the infimum over all subordinate norms of the corresponding norm of the operator.

I am trying to understand a proof I have seen of the following theorem: $$\rho(A)=\inf_{\|\cdot\|}\|A\|.$$ I understand that to do this, the idea is to show that 1) $\rho(A)\leq\|A\|$ for any norm, ...
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89 views

Operator norm of the inverse

If I made no mistake, one can calculate the operator norm of the inverse of any given (invertible) operator $A: V\rightarrow V$ via: \begin{align}\|A^{-1}\| & = ...
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54 views

Minimize Function over Convex Subset

Suppose that C is a closed convex subset of $\mathbb R^n$ and $x \in \mathbb R^n$. The projection of $\mathbf x$ onto C is the closest point $\mathbf y \in C : \mathbf z = \mathbf y$ minimizes ...
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71 views

How to FAST calculate 2 norm / spectral norm of a matrix.

I meant reduced 2 norm, the largest singular value. My current approach is applying the SVD decomposition of A via "?gesdd" in MKL, and then taking the largest singular value. I think there should ...
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52 views

Equivalence of norms proof

This question is from a set of optional, much harder problems from my first year analysis course, but the subject material is norms on $\mathbb R^K$. (c) Show that there exists a constant $C > 0$ ...
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37 views

Does the closed unit ball of $C(E)$ have no extreme points?

Let $E$ be a bounded closed set in $\mathbb R^n$. Does the closed unit ball of $C(E)$ (the space of continuous functions on $E$ with supremum norm) have no extreme points?
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27 views

Which of the following expresses the fact that the vectors $u$ and $v$ have the same length?

Which of the following expresses the fact that the vectors $u$ and $v$ have the same length? (a) $u · u = v · v$ (b) $||u + v|| = ||u|| − ||v||$ (c) ${u \over||u||} = {v \over ||v||}$ (d) $||u + ...
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Matrix - calculate norms

Let be a matrix of $n \times n$ elements. It has on each column $k$ elements equal to $1$ and all the rest equal to $0$. The question is : calculate the minimum and maximum norms $\|A\|_p$ of this ...
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31 views

Find conditioning of the matrix

Find conditioning of the following matrix: $$A=\begin{bmatrix}1& 0\\1&\epsilon\end{bmatrix}.$$ in a $\|.\|_\infty$ norm for $\epsilon > 0$
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Immediate consequence of the definition of Operator Norm. Explain

||Av|| $\leq$ ||A||$_{op}$||v|| for every v $\in$ V I was wondering why this is true. Wikipedia says it's an immediate consequence of the definition but I just do not follow. I am using the ...
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38 views

Is $u^TAu \geq 0$ true for all symmetric matrices $A$?

we know from the definition of inner product and norm, that $u^Tu$ is always larger than zero, except the case where $u=0$ at which case it is zero. I came across a question that infers that $u^TAu ...
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Positivity of a function in $\mathbb{R}^{n}$

We place ourself in $\mathbb{R}^{n}$. We consider a given increasing function $$ g : \begin{aligned} &\mathbb{R}^{+} \to \mathbb{R} \\ &x \;\;\,\mapsto g(x) \end{aligned}$$ Finally, we ...
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How to prove this inequality for operator and function

How to prove this? $\sum_{k=1}^{\infty}|(Tf)_k|^2\leq ||T||^2||f||^2$ where $T$ is an operator and a function $f$. $(Tf)_k$ is the $k$-th coordinate of Tf. Should this involve Cauchy-Schwarz or the ...
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Frobenius norm bound

Is there any way to bound Frobenius norm of a product of square matrices A,B and a vector x in the following way: $$ \|ABx\|≤ \|Ax\|\text{ and }\|B\| $$
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29 views

Does this question make any sense - eigenvalues and norms

Im having difficulties understanding this question: show that if $b$ is an eigenvector of an invertible matrix $A$ with an eigenvalue $\lambda_1$ and $\delta b$ is an eigenvector of $A$ with an ...
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49 views

proving uniform convergence for a uniformly continuous function

I've been stuck on this question (2a) for about a day now! Question and havn't really gotten anywhere, I think that the series is that integral of a step function that approximates f ie: $\Gamma_n ...
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34 views

Prove triangle inequality of vector norm

I am trying to show that $||x+y||_p \leq ||x||_p + ||y||_p$ where $p$ is an integer larger than 1, but not infinity (I proved those cases already), and $||x||_p = (\sum_{i=1}^n |x_i|^p)^{\frac{1}{p}}$ ...
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72 views

Consistency of matrix norm: $||Ax||_2 \leq ||A||_{Frobenius}||x||_2$

I'm trying to show that $||Ax||_2 \leq ||A||_{F}||x||_2$ where $A$ is an n by n matrix, $x\in \mathbb R^n$, $||x||_2$ is the euclidean norm, and $||A||_F$ is the frobenius norm. I actually wrote ...
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Spectral Radius and Norm of multiplied vector

Let $\mathbf{A}$, $\mathbf{B}$ be square matrices of equal dimensions, $\mathbf{w}$ a vector of compatible dimensions and $\rho$ be the spectral radius operator. Does the following hold? If $\rho ...
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Matrix Norm Lemma

There is a lemma claims that : $||Ax||/||x|| \le max_{||x||\ne 0} (||Ax||/||x|) = ||A|| $ I'd like to know how come $||Ax||/||x|| \le max_{||x||\ne 0} (||Ax||/||x|)$ because it does not make sense ...
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57 views

Triangle inequality frobenius norm

I'm trying to show that the frobenius norm is a norm. however it appears as if triangle inequality isnt met. $$||A+B||_F = \sqrt{\sum_{i,j=1}^n |a_{ij}+b_{ij}|^2} \leq \sqrt{\sum_{i,j=1}^n ...
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132 views

Derivative of a matrix: Outer product chain rule

I ran into a seemingly simple matrix calculus question that I can't seem to find the solution to. Suppose I have the following matrices: $X_{(t \times n)}, V_{(n \times m)}$, and $\Phi_{(t\times m)} ...
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38 views

l1 minimization with orthogonality constraint

I want to find a rotation (or reflection) for my data which maximizes the space between my points and the basis' margins. I have formulated the problem as follows: Given $X \in \mathbb{R}^{n \times ...
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84 views

Minimize norm - Least Squares - Linear Algebra

Given $Ax = b$, I know how to use least squares to minimize $||Ax-b||^2$. How do I minimize the 2-norm ($||x||_2$) and the Frobenius norm of $x$?
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60 views

Maximum Norm versus Euclidean Norm of Projection Difference

OK, there's more to it but I couldn't fit everything in the Title. This is the situation: I have a subspace of $R^n$, call it $I$, which is contained in $Z \equiv R^n \cap \{x : \sum_i x_i = 0 \}$ ...
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49 views

Matrix-Vector Product

Suppose I have the expression $\lVert\mathbf B \cdot\hat n\lVert=1$, where $\mathbf B$ is a matrix and $\hat n$ is a unit vector (both can have any dimensions, as long as they are compatible). What ...
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69 views

Is is true that $||v+w||^2 = ||v||^2 + 2\langle v,w \rangle + ||w||^2$?

Is it true that for a $\mathbb{R}$ vector space with dot product $\langle\cdot, \cdot\rangle$ and $||\cdot||$ norm \begin{align} ||v+w||^2 = ||v||^2 + 2\langle v,w\rangle + ||w||^2 && ...
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34 views

Norm on space of test functions.

What is $\nabla^{j}f(x)$ for $f:\mathbb{R}^{n}\rightarrow{\mathbb{C}}$ in this note which is just after Exercise 1? It is mentioned there that is $d^{j}$-dimensional vector but I am not able to get ...