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

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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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Norm of matrix exponential

If $$\phi(t,0) = \exp(At)$$ and $$\|\phi\|<\exp(a+bt),$$ how to find the values of $a$ and $b$ (using equations)?
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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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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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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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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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41 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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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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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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Dose the closed unit ball of C(the closure of E) with sub-norm, have no extreme points?

Let E be a bounded closed set in R^n. Dose the closed unit ball of C(E) with sup-norm, have no extreme points?
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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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26 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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37 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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L1 norm of a trigonometric polynomial

For a real $x$, $f(x) = \sum_{k=-T}^{T}e^{ikx}$ is the well known Dirichlet kernel. It is also known that $\|f\|_{L_1}=\int|f(x)|dx \le C_1\log T + C_2$ for some $C_1,C_2$ independent of $T$. ...
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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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24 views

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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25 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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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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24 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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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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34 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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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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26 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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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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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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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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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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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 ...
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gradient norm of a simple function

In this answer Derivation of soft thresholding operator how can I derive that $\nabla(||x-b||_2^2)=b-x$?
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Total Variation minimization problem

Thanks for reading this thread. I have a object function, with constraints, I am trying to minimize. The object function is the Total Variation of an image. The Total Variation is defined as: ...
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Norm inequality with wedge product

Anyone could help me to prove this following inequality? $\displaystyle\frac{||(u+v)\wedge w||}{||u+v||}\le \frac{||u\wedge w||}{||u||} +\frac{||v\wedge w||}{||v||} $ where $u\wedge v$ is the wedge ...
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Equivalence of Frobenius norm and trace norm

According to [1], [2] and other related publications, the following holds for any matrix $X$: $$\| ...
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Show that $\lim_{a \to \infty} \sup_{n} \int_0^{T-a}||v_{n,r}(t+a)-v_{n,r}(t)||_{\mathbb{L}^2(\Omega_{2r})}^2dt=0.$

Let $\ 0 \leq t \leq t+a \leq T$, with $$\lim_{a \to 0} \sup_{n} \int_0^{T-a}\left\|u_n(t+a)-u_n(t)\right\|_{\mathbb{L}^2(\Omega_{r})}^2dt=0,$$ where $\Omega_r=\Omega \cap \left\{x \in \mathbb{R}^2; ...
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Uniformly Bounded in a Schauder Norm

Suppose $\{h_{k}\}$ is a sequence of elements in $C^{2,\alpha}$. I want to show that $\{h_{k}\}$ is uniformly bounded in $C^{2,\alpha}$. Now, if I'm not mistaken, then $||h_{k}||_{\displaystyle ...
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Completeness is not preserved under homeomorphism

I am trying to give a counterexample which will make it clear that homeomorphism does not preserve completeness. I have an easy one in mind ($(0,1)$ and $\mathbb{R}$) but I have just thought that ...
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Restrictions on a Matrix-Vector product

Suppose I have a $m\times n$ matrix $\mathbf M$, and a unit vector $\hat v$, of dimension $n$. What restrictions do I need to apply to $\mathbf M$ so that $\lVert \mathbf M\cdot \hat v\lVert \leq 1$ ...
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multivariable calculus question norm

Given vector space C([a,b],$ \mathbb{R} $) of continuous functions of [a,b] in $ \mathbb{R}. $ Prove that the function $ \left \| f \right \|_{1}=\int_{a}^{b}\left | f(t) \right |dt $ is a norm. Also ...
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Check continuity and find norm of linear functional

I have to check continuity and find norm of the following linear functional $$\ell^{2}\ni \{x_n\}^{\infty}_{n=0} \rightarrow \sum^\infty_{n=0}\frac{x_n}{\sqrt{2^n}}\in \mathbb{K}$$ As for continuity, ...
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Normed linear space with two norms that are not equivalent, one is complete, what about the other?

I have been searching for an answer to the following question: Given a normed linear space $V$ and two norms that are not equivalent, but $\exists K\in\mathbf{R}$ such that $\|v\|_1\leq K\|v\|_2$ ...
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Multivariable calculus - find total derivative

I want to find the total derivative of the function $f: \mathbb R^n \to \mathbb R^n$, $f(x)=\frac{x}{|x|}$ If I was to copy what the teacher taught, I should find the limit of $\lim_{t \to 0} ...
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Topological equivalence of any norm on $\mathbb C^n$

In University I have been told that every norm on $\mathbb C^n$, for any $n\in\mathbb{N}$, is equivalent to every other such norm. I have a proof for this on any vector space on $\mathbb R$. Trouble ...
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Asymptotics of Gelfand's formula

In the following paper, it is stated that for for any matrix norm, $n \in \mathbb{N}$ and $A \in \mathbb{C}^{d \times d}$, the following holds: $\rho(A) \ge \gamma^{(1+\ln n)/n}\|A^{n}\|^{1/n}$ for ...
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Sobolev Inequality with powers

The following is one of the Sobolev Inequalities as stated in Gilbarg & Trudinger's text on elliptic PDEs: $\displaystyle W_{0}^{1,p}(\Omega)\subset L^{\frac{np}{(n-p)}}(\Omega)$ for $p<n$. ...
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Show if $||\cdot||$ is a norm on $\mathbb{R}^m$ and $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is linear and one to one the following is a norm.

Show if $||\cdot||$ is a norm on $\mathbb{R}^m$ and $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is linear and one to one then $||\cdot||_*: \mathbb{R}^n \rightarrow \mathbb{R}$ given by $||x||_* = ...
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About matrix products $A^{T}A$ and $ AA^{T} $

I'm investigating the relationship between 2-norms and eigenvalues of $A^{T}A$ and $ AA^{T} $, in order to better understand the SVD decomposition. How can I prove that $A^{T}A$ and $ AA^{T} $ are ...