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

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Subordinate matrix norm

I have the following matrix norm: $$\Vert A \Vert = \max_{1\leq i, j\leq n} \vert a_{ij} \vert \>.$$ I have to decide if this is a subordinate matrix norm or not. I have tried to use the ...
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If two norms are not equivalent, then we have a sequence $\|x_n\|/\|x_n\|' \to 0$.

Definition 2.22. We say that two norms $\|\cdot\|$ and $\|\cdot\|'$ are equivalent if there exist $C_1,C_2>0$ such that $$C_1\|x_1\|' \le \|x\| \le C_2 \|x\|',$$ for all $x\in V$. If two ...
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Equivalence of complete norms

Let $\|.\|_1$ and $\|.\|_2$ be two complete norms on a linear space $X$ such that if a sequence $(x_n)$ converges to $x$ in $(X,\|.\|_1)$ and to $y$ in $(X,\|.\|_2)$, then $x=y$. We have to prove that ...
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Prove that there exists NO norm such that $ f_n $ converges to $f$ iff $f_n$ converges to $f$ on compacta.

More specifically: Given the space of continuous functions $\mathbb{R}\rightarrow\mathbb{R}$ where convergence is equivalent to uniform convergence on compacta (i.e., compact sets), prove that ...
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Prove norm inequality

It is given that $$\left\lVert x-y\right\rVert =\left\lVert y-z\right\rVert = \left\lVert z-x\right\rVert \qquad (1) $$ where $x,y,z \in \Bbb R^2$ and $ \left\lVert x\right\rVert=\sqrt ...
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40 views

Interchange of $\ell^r$ and $L^p$-norm

Let $(f_i)_{i\in\mathbb{N}}$ be a sequence of $L^p$-functions. What is the relation between $\Vert \Vert (f_i)_{i\in\mathbb{N}}\Vert_{\ell^r}\Vert_{L^p}$ and $\Vert \left(\Vert ...
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Derivative of a nonsingular matrix

Show that : $$\frac{d}{dt} A^{-1}(t) = -A^{-1}(t) (\frac{d}{dt} A(t) ) A^{-1}(t) $$ A(t) is a matrix.
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Why “Re” when squaring norms?

Why does the "Re"-operator pop up in equations when we take a norm and square it? Take, for example, $$\| h - u \|^2 + \|h - w \|^2 = Re\langle h-w, w - u \rangle + \|u - w\|^2 $$ which is taken from ...
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Linear functions $\mathbb{C}^n\longrightarrow\mathbb{C}^m$ are Lipschitz continous

Exercise: Show that any linear function from $\mathbb{C}^n$ to $\mathbb{C}^m$ is Lipschitz continous. (Hint: Use suitable norms.) I know that the maximum norm, the euclidean norm and the sum norm ...
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The norm of the extension of an operator

If $T_0$ is the identity operator from $E$ to $E$, where $E$ is a subspace of $F$ and both of them are Banach spaces (maybe not needed). If the bounded linear operator $T$ is an extension of $T_0$ ...
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Calculating $A^T A$ in matrix with orthogonal columns

I have a matrix $A$ with three orthogonal columns, and I know that the length (2-norm) of each column is $4$. The question is: what is $A^T A$? Which properties should I use to solve this? Thanks ...
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Prove continuity for a given norm

I struggle with this exercise from an analysis 2 book I use for self study: Let V := $C^1([0,1]; \mathbb{C})$ the vector space of continously differentiable functions from $[0,1]$ to $\mathbb{C}$ ...
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29 views

Why can the function $f(x)=||A\vec{x}-\vec{b}||^2$ be rewritten as $\vec{x}^tA^tA\vec{x}−\vec{x}^tA^t\vec{b}−\vec{b}^tA\vec{x}+||\vec{b}||^2$

Someone answered a question introducing this transformation of the function $f(x)=||A\vec{x}-\vec{b}||^2$ ; but I cannot get the idea why and how. Looks a bit like a binomial expansion, but I can't ...
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48 views

Question about an inequality in a published paper which is yielded from an approximation

I am reading a published paper on K- SVD: An algorithm for designing overcomplete dictionaries for sparse representation In the introduction, it says: Recent years have witnessed a growing ...
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Squaring Norms solved by Algebra

I found the following in a paper and am not sure how it is correct: $\Vert a - b \Vert^2$ was expanded to: $\Vert a \Vert^2 - 2a^Tb + \Vert b \Vert^2$ The paper was on gps location algorithms so ...
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Prove that $(\|x\|^p_X + \|y\|^p_Y)^{1/p}$ is a norm

Let $X$ and $Y$ be normed spaces equipped with the norms $\|\cdot\|_X$ and $\|\cdot\|_Y$, then prove that the following defines a norm on $X\times Y$ for $1\le p < \infty$: $\|(x,y)\| := ...
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Prove there cannot be an inner product which turns $l^p$ into an inner product space?

For all $1\leq p < \infty, \mbox{ }p$ is not equal to 2, prove there cannot exist an inner product that turns $(X,\|\cdot \|_p)$ into an inner product space; that is, prove that there cannot be ...
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436 views

a question about symmetric positive definite matrix and norm

If B is $n\times n$ real symmetric positive definite matrix, then $(x,y)=x^TBy$ definites an inner product on $R^n$. How to prove that $||x||=(x^TBx)^{1/2}$ is a norm on $R^n$?
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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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Any two points inside a circle are within a diameter of each other.

In many problems involving the Pigeonhole Principle, we often assume the following lemma: Lemma: The distance between any two points in a circle of radius $r$ is at most $2r$. Intuitively, this ...
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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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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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Submultiplicativity stronger than triangle inequality?

I would like to ask a question about matrix norm. Is the submiltiplicativity property always stronger than the triangle inequality? So, if i prove for a matrix norm that it's submultiplicative, i ...
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Parallelogram law in normed vectorspace withour an inner product.

Let $V$ be any $\mathbb{K}$-vectorspace with norm $\|\cdot\|$ I know that the Parallelogram law holds if the norm is induced by some inner product $\langle\cdot,\cdot\rangle$, i.e. $$ ...
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Must vectors in $\mathbb{R}^n$ have their “tail” at origin?

I was looking the definition for an $n$-sphere centered at origin with radius $r$: $$\mathbb{S}^n = \{v \in \mathbb{R}^{n+1} : ||v|| = r \}$$ Although I understand that the $||v|| = r$ condition ...
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Is there any matrix norm $|| \cdot ||$ such that $||A|| \le ||A||_{\infty} /n$?

Matrix norms are equivalent and can bound each other like some examples on Wikipedia. I was wondering if there is a matrix norm $|| \cdot ||$ that can be upper bounded by $||\cdot ||_{\infty}/n$ ? ...
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Prove that the square sum of eigenvalues is no more than the Frobenius norm for a square complex matrix

Prove: $$ \sum_{r=1}^{n} |\lambda_r|^2 \le \sum_{i,j=1}^{n} |a_{ij}|^2 $$ the equality holds if and only if $\boldsymbol{A^H A=AA^H} $ for a square complex matrix $ ...
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Prove that $d_{1}$ and $d_{2}$ induce the same topology

Suppose $d_{1}(x,y) = |x-y|$, $d_{2}(x, y) = |\phi(x) - \phi(y)|$, where $\phi(x) = \frac{x}{1 + |x|}$. Prove that $d_{1}$ and $d_{2}$ are metrics on $\mathbb{R}$ which induce the same topology. ...
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Triangle inequality in product space of normed spaces

Let $(X,||.||_X)$ and $(Y,||.||_Y)$ be normed spaces, then $||(x,y)||:=(||x||_X^p+||y||_Y^p)^{\frac{1}{p}}$ is a norm on $X \times Y$. This is absolutely clear to me, but I have troubles to verify ...
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In a normed vector space $(V,\lvert . \rvert)$ show that $f:V\rightarrow \mathbb{R}$ with $f(v)=\lvert v\rvert$ is uniformly continuous

In a normed vector space $(V,\lvert . \rvert)$ show that $f:V\rightarrow \mathbb{R}$ with $f(v)=\lvert v\rvert$ is uniformly continuous The first part of the question says to prove the "reverse ...
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$\lambda_k \to 0$ implies $T$ is compact?

I am doing an exercise which asks to show that if $\{\varphi_k\}$ is an orthonormal basis in a Hilbert space with $T$ a bounded operator satisfying $T\varphi_k = \lambda_k \varphi_k$, then $\lambda_k ...
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A question about a proof for why $\|x\|:=\inf\{\lambda>0\mid\frac{x}{\lambda}\in B\}$ is a norm

I started studying functional analysis, a claim that was thought is the second lecture claims that: Let $X$ be a vector space, $B\subseteq X$ is convex, symmetric around $0$ and s.t ...
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102 views

Is any norm on $\mathbb R^n$ invariant with respect to componentwise absolute value?

Given $\mathbf{x}=(x_1,...,x_n) \in \mathbb{R}^n$ , define $ \mathbf{x}'=(|x_1|,...,|x_n|) $ . Then, is it $||\mathbf{x}'|| = ||\mathbf{x}||$ for every norm on $ \mathbb{R}^n $ ? NB: The answer ...
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Inequality for norms

Let g(x, y) be function on $X\times Y$. Show that for all $p\geq q$ $$ \|\,\|g\|_{L^q(Y)}\,\|_{L^p(X)}\leq \|\,\|g\|_{L^p(X)}\,\|_{L^q(Y)} $$ Thsnk you.
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Clarification with a identity

I have the following inequality which comes in proof of triangle inequality. $\|a+b\|^2=\|a\|^2$+$\|b\|^2+2\Re\langle a|b\rangle\le\|a\|^2+\|b\|^2+2|\langle a|b\rangle |$ I don't know where the ...
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Norms Abstract Analysis

I have a question relating to norms and have been giving functions and need to state whether they are norms or not... which of the following are norms on $\mathbb{R}^2$? Give reasons for your ...
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equivalence of norms and direct sum

Let $(X,\|\cdot\|_X) $ be an infinite dimensional Banach space. Suppose that you can write $X=V\oplus W$. Write $x=v+w$ and define $(V\oplus W,\|x\|_\oplus :=\|v\|_X+\|w\|_X)$. Show that ...
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Distance between point and linear Space

Suppose $E$ is a normed vector space. Let $f$ be a continuous linear functional on $E$ and denote by $M$ the Kernel of $f$. Let $x\in E$. How to show that ...
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Determine operator norm of mutiplication operator

Consider $$T: (C[-1,1],\|\cdot\|_{2})\rightarrow \mathbb{C}\\Tf :=\int_{-1}^{1}mf\,\mathrm{d}x$$ where $m\in C[-1,1]$. I want to prove $\|T\| = \|m\|_2$. $\|T\|\leq\|m\|_2$ can be easily proved by ...
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proving $L^\infty$ norm inequality (disprove $\Vert f\Vert_\infty\le\sqrt{n}$)

There are three parts in this question, I've done the first two but not sure about the third one. Also see $L^2$ norm inequality. In the third part, I am asked to show that if $W$ is a ...
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How to prove positive definiteness?

$B_{(n+1)(n+1)}$ = $ \begin{bmatrix} A & u \\ u^T & 1 \\ \end{bmatrix}$ is given, and $A$ is a positive definite matrix where its Cholesky factorization is given ...
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Norm and continuous functions

Having the definition: A function $f:\mathbb{R^n}\rightarrow \mathbb{R^n}$ is proper if $\|f(x)\|$ tends to $\infty$ when $\|x\|$ tends to $\infty$. I have to show : a)If $f:\mathbb{R^n}\rightarrow ...
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Infinity matrix norm example

I have a brief question regarding the infinity matrix norm. The subordinate matrix infinity norm is defined as: $$\|A\|_{\infty} =\max_{1 \leq i \leq n}\sum_{j=1}^{n}|a_{ij}|.$$ This is derived ...
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Proving norm inequality

There is a brief proof in my textbook that I have one question about. We are supposed to prove that $||x||_{1} \leq n||x||_{\infty}$ for $x \in \mathbb{R}^n$ The book writes the following: ...
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Limit inferior taken on the norm of a sequence

Let $E$ a normed vector space and let $(x_n)$ be a sequence in $E$. Suppose that $x_n$ converges weakly (i.e. wrt the weak topology) to $x$. Why is it that from the inequality $$ |f(x_n)| \leq \|f\| ...
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Why one would want to normalize a matrix by dividing it by its Frobenius norm?

I am currently reading a scientific paper about clustering of brain signals, which consist on long time series across many channels (each signal is a matrix of C channels by T time samples). In the ...
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Do real vectors attain matrix norms?

The $p$-norm of a matrix $A$ for $p \geq 1$ is defined as $$\|A\|_p = \max_{ x \in \mathbb{R}^n, \|x\|_p=1} \|Ax\|_p.$$ My question: does this equal $$ \max_{ x \in \mathbb{C}^n, \|x\|_p=1} ...
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Norm of the linear bounded operator $T$ defined by $(Tf)(x) = \int_0^x g(t)f(t)dt$

Some time ago my teacher showed the solution of this exercise. Today I reviewed it, and I think he might be wrong at the last part, c.) Exercise: Let $a > 0$ and let $g \in C[0,a]$ be a ...
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Prove that $\max_{|z| = 1} |P(z)| \ge 1$

I got stuck on this problem: Given a polynomial on complex plane $P(z) = z^n + a_{n-1}z^{n-1} + ... + a_1 z + a_0$ for $z \in \mathbb{C}$. Prove that $\max_{|z| = 1} |P(z)| \ge 1$ What I tried ...
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Verify that $\| T(x) \| = \| x \|$

Let $T:\mathbb{R}^3 \rightarrow \mathbb{R}^3$ be multiplication by the matrix $$A= \begin{bmatrix} \frac{1}{3} & \frac{2}{3} & \frac{2}{3} \\[0.3em] \frac{2}{3} & -\frac{2}{3} & ...