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

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Equivalence of given norms in $\mathbb{R}^k$

Find as big as possible $c>0$ and as small as possible $C>0$ such that we have: $$\forall_{\vec{x}\in\mathbb{R}^k} \ \ c\cdot \sqrt{\sum_{j=1}^k|x_j|^2}\le \sqrt{\sum_{j=1}^k ...
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27 views

Why is the tensor product of admissible homomorphisms of seminormed A-modules admissible?

Let $A$ be a normed ring, and let $M$ and $N$ be seminormed $A$-modules. If $\phi:M'\rightarrow M$ and $\psi:N'\rightarrow N$ are admissible homomorphisms of seminormed $A$-modules, then I want to ...
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175 views

Gradient of a norm with a linear operator

In mathematical image processing many algorithms are stated as an optimization problem, where we have an observation $f$ and want recover an image $u$ that minimizes a objective function. Further, to ...
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73 views

Norm of inverse, Banach algebra, Proof of Gelfand-Mazur theorem

In a proof of the Gelfand-Mazur theorem, I read that $$ \lim_{\lambda\to\infty}\|(a-\lambda 1)^{-1}\|=0 $$ (where it is assumed for the sake of contradiction that the inverses in the norm exist for ...
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3answers
70 views

Compute the norm of matrix

Let $M$ be $n\times n$ matrix, consisting entirely of 1's. Show, that $\|M\|_{op}=\sup_{x\in C^n}|Mx|=n$.
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134 views

Spectral radius, second induced norm

In my textbook there are few facts left without any sign of a proof, which really bugs me, and I was thinking maybe someone can help me: $A\in \mathbb{R}^{m,n} \Rightarrow \ \|A\|_2 = ...
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60 views

Norm of a bounded operator

Let $\phi \in C[0,1]$ and $T_{\phi}:C[0,1] \rightarrow \mathbb{R}$ be: $T_{\phi}f = \int_0^1 f(x)\phi(x)dx$ prove that $T_{\phi}$ is a continuous, linear functional and that $||{T_{\phi}}|| = ...
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54 views

Relation of norms of matrices

Let $A$ be $m \times n$ matrix. Let $B=\frac 1n AA^*$, where $A^*$ is a transposed matrix. Let $X_i, I\leq m$ be row-vectors of $A$. Show $$ \|B\|=\frac 1n \|A\|^2\geq \max_{i\leq m}|X_i|, $$ Where, ...
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27 views

Normalized cross-correlation in detail

I'm trying to implement a normalized cross-correlation algorithm but I don't get what in fact is this measure. What confuses is the wikipedia definition: $\frac{1}{n} \sum \frac{(f(x,y)- ...
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166 views

Inequality for Euclidean norm

Let:| | be Euclidean norm on $\mathbb{R}^{n}$ and $b : \mathbb{R}^{n}\longmapsto \mathbb{R}^{n}$ and $\sigma : \mathbb{R}^{n}\longmapsto \mathbb{R}^{n\times m}$ two continuous functions. ...
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278 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 ...
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472 views

Why does spectral norm equal the largest singular value?

This may be a trivial question yet I was unable to find an answer: $$\left \| A \right \| _2=\sqrt{\lambda_{\text{max}}(A^{^*}A)}=\sigma_{\text{max}}(A)$$ where the spectral norm $\left \| A \right ...
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89 views

Matrix norm equivalence

If we define $ \|A\| = \max \{|A\cdot \mathbf{t}|:|\mathbf{t}|\leq 1\}.$ is it the same as defining it as $\max \{|A\cdot \mathbf{t}|:|\mathbf{t}|= 1\}$ ? If so, why? The book I'm following uses the ...
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53 views

Show that every operator norm is consistent

Is the following a correct way to show that operator norms are consistent? $$ \|AB\|=\max_{Bx \ne 0}\frac{\|ABx\|_\alpha }{\|x\|_\alpha} =\max_{ Bx\ne 0}\frac{\|ABx\|_\alpha}{\|Bx\|_\alpha} ...
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89 views

Abstract Algebra: Field extensions

I'm trying to prove the following but it causes me a lot of trouble: Let $L$ be a finite extension of degree $n$ of a field $K$ with characteristic $0$. Let $\sigma_1,\dots,\sigma_n$ be different $K$ ...
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168 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 ...
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61 views

continuity of $L^p$ norms with respect to $p$

Let $0<p_0<p<p<p_1\leq \infty$. Then I have proved $L^{p_0}(\mu)\cap L^{p_1}(\mu)\subseteq L^{p}(\mu)$. In particular, when $p_0=1$, $p_1=\infty$, I have proved further ...
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37 views

When is $c v^\top y y^\top v \ge ||v||^2$

Given $c\in R$ being some constant, $v, y \in R^n$, I want to find conditions for which the following inequality holds true: $$c v^\top y y^\top v \ge ||v||^2$$ EDIT: Note that $y y^\top$ is an $n ...
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46 views

Norm of operator $T_x(f) = f(x)$

Let $X$ be a normed vectorspace and $X'$ be the dual space of $X$. For $x \in X$ we can define $T_x: X' \to \mathbb F$ by $T_x(f) := f(x)$. This is indeed an operator in $X''$. I read that $\| T_x \| ...
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71 views

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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75 views

There exists function sequence $\{f_{n}\}$ converges to $0$ such that $\{a_{n}f_{n}\}$ not converges to $0$

Let $X$ be the vector space of all complex functions on the unit interval $[0,1]$, topologized by the family of seminorms $$p_{x}(f) = |f(x)|, \quad (0 \le x \le 1).$$ Show that there exists a ...
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60 views

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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136 views

L2 Norm of Gaussian Integer

I'm writing a Java program that deals with Gaussian Integers. In my program, I have to compute the L2 norm of the GI (Gaussian Integer) and return it as a float. I've looked around but I cannot seem ...
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Minkowski type inequality in Banach algebras

Under which circumstances it is true that $\|(A+B)^n\|^{1/n}\le \|A^n\|^{1/n}+\|B^n\|^{1/n}$ for elements $A$ and $B$ in a Banach algebra and a natural number $n$?
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44 views

Can we write $||A - B|| \leq ||A||$?

I am confused with the very basic question related with the matrix norm. Can we write $||A - B|| \leq ||A||$ ? Thanks for the help and time.
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39 views

Norm of a vector is determined by evaluation of linear functionals on it: can this be proved without the Hahn-Banach theorem?

Let $V$ be a normed vector space over the field of real numbers, $\mathbb R$, and let $x_0 \in V$ be fixed. I know how to prove $$\|x_0\| = \sup_{f \in V^*, \|f\| = 1} |f(x_0)|$$ using the Hahn-Banach ...
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For which $L^p$ is $\pi=3.2$?

This is a silly question that came to mind after watching numberphile video on How Pi was nearly changed to 3.2. For which $p\in(1,+\infty)$ is the ratio of the perimeter of the $L^p$ disc in $\Bbb ...
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65 views

Prove that $|x(a)| + \max \{|x'(t)|: t \in [a,b]\}$ is a norm for a complete space

Prove that the space $C^1([a,b])$ consisting of continuous functions in $[a, b]$ with the norm $|x(a)| + \max \{|x'(t)|: t \in [a,b]\}$ is a Banach space. I can't prove the completeness of this ...
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133 views

Calculate the derivative of a complex norm

I'm stuck with a rather trivial looking question. How do you calculate the derivative of the norm of a complex number to it self? Like in $$ \frac{d|a|^2}{da} = ? $$ I think it would give rise to a ...
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65 views

Find norm of a linear functional

I have a normed space: $$l_p = \{ (x_n)_{n = 1}^{\infty}: \sum\limits_{n = 1}^{\infty}|x_n|^{p} < \infty \}$$ With norm: $$||x|| = (\sum\limits_{n = 1}^{\infty} |x_n|^p)^{1/p}$$ where $p = ...
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Discontinuous function

Show that the function, $f:\mathbb{R^2}\rightarrow \mathbb{R}$, $$ f(x,y) = \begin{cases} 1, & \|(x,y)\|\geq1\\ 0, & \|(x,y)\|<1 \end{cases} $$ is not continuous ($\mathbb{R}$ and ...
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Calculate $\|f\|$ with $f(x) = \int_{-1}^{1}x(t)dt - 2x(0)$

In $C[-1,1]$, consider the norm $\|x\| = \max\{x(t)\mid t \in [-1,1]\}$ and $f(x) = \int_{-1}^{1}x(t)dt - 2x(0)$, $x \in C[-1,1]$. Find $\|f\|$ Hi everybody. I got stuck in this problem. I can ...
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57 views

What is the norm of v + w

So the norm of v is defined as ||v|| = sqrt(v_1^2+...+v_n^2) I would guess that the norm of v+w is defined as ||v+w|| = sqrt((v_1+w_1)^2+...+(v_n+w_n)^2) Is this correct? My textbook asks me to ...
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52 views

inequality about Inner product and norm

If $m\times n~(m<n)$ matrix $A$ satisfy the following condition $(1-\delta)||s||_2^2\leqslant \|As\|^2_2\leqslant (1+\delta)\|s\|_2^2$ for all the $n \times 1$ vector with no more than $k$ nonzero ...
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51 views

calculating norm of the following matrix?

I have the following matrix and its norm $$ \begin{aligned} A(s) &= \begin{bmatrix} 0 & e^{-sT} \\ e^{-sT} & 0 \end{bmatrix} \\ \lvert \lvert {A(s)} \rvert \rvert_{\infty} &= ...
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50 views

Norm of a linear mapping, please check if what I have done is right

please check if what I have done is right. $C[0,1]=$ continuous functions in $[0,1]$ considering $\|g\|=\max_t|g(t)|$ $$X=\langle t^2,1 \rangle $$ the subspace of $C[0,1]$ generated by $t^2$ and $1$ ...
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120 views

Strange mistake in this proof about normed separable vector spaces.

I am supposed to prove that an infinite-dimensionale separable normed space $X$ constains a countable subset $Y$ that is linearly independent and dense. There are three hints given: First, prove ...
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$|Av||A^{-1}v|$, $A$ non-singular, $|v|=1$.

Let $A$ a non-singular $n \times n$ matrix, $v \in \mathbb{R}^{n}$ a variable vector. The operator norm of $A$ is defined to be $|| A ||=\max_{|v|=1} |Av|$ where $|Av|$ is the standard Euclidean norm ...
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59 views

Frobenius Norm Unitary Operators

For something I'm working on, I have a matrix $A$ with other matrices $U$ and $V$ which are unitary ($U^*U = I$ and $V^*V = I$), and I'm trying to show that, for the Frobenius norm, $\|A\| =\|UA\| ...
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45 views

Inequality of scalar-product and norm

Why does the following inequality hold, given $A$ is symmetric and $\lambda_{\min} (A)$ is the smallest Eigenvalue of $A$? $$v^\top A v \ge \lambda_{\min} (A) \; ||v||^2$$
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20 views

Following positive semi-definitness from matrix norm

How can I follow the following? $$||A||_2 \le \sigma > 0$$ $$\Leftrightarrow A -\sigma I \mbox{ is positive semi-definit}$$ I always get it the other way around, i.e. that $\sigma I - A$ is ...
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1k views

Frobenius Norm Triangle Inequality

How can I go about proving the triangle inequality holds for the Frobenius norm? I worked through $\|A+B\|_F \le \|A\|_F + \|B\|_F$ and was not able to make it work =/.
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417 views

Frobenius Norm Properties

Say $v\in {\mathbb{R}^{l}}$ (a column vector) and $A\in {\mathbb{R}^{l\times l}}$. I have encountered the following equality: $||v{{v}^{T}}-A||_{F}^{2}={{\left( {{v}^{T}}v ...
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34 views

Integral like a norm instead of sum

A Rienmann integral is defined as: $\int_a^b f(x)\ dx=\displaystyle\lim_{n\rightarrow\infty}\displaystyle\frac{1}{n}\displaystyle\sum_{i=0}^{n}f\left(\displaystyle\frac{b-a}{n}\cdot i+a\right)$ I ...
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88 views

Frobenius Norm with Unitary Operators

For something I'm working on, I have a matrix $A$ with another matrix $U$ which is unitary ($U^*U = I$), and I'm trying to show that, for the Frobenius norm, $\|A\| =\|UA\|$. Now, I can do this ...
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2answers
42 views

If $||f-g|| < ||f^{-1}||^{-1}$, then $f$ is isomorphism implies $g$ is also isomorphism

Let $E$, $F$ be Banach space, $f,g \in L(E,F)$ and $f$ is isomorphism. Prove that if $||f-g|| < ||f^{-1}||^{-1}$, then $g$ is isomorphism. Hi everybody. I got stuck on this problem and can't ...
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82 views

Plot of ||X||infinity norm

Can anybody tell me why the plot of $\|X\|_{\infty}$ in $\mathbb{R}^2$ comes out to be square? Since $\|(x_1,x_2)\|_{\infty} = \max\{|x_1|,|x_2|\}$, then let us say $|x_1|$ is max. Why the plot is ...
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1answer
42 views

A normed space of continuous functions with norm $\int_{0}^{1}|f(t)|dt$ is not complete

Suppose $E$ is a normed space of all continuous functions on $[0,1]$ with norm $\int_{0}^{1}|f(t)|dt$. Prove that $E$ is not complete I know that we must do is to find a Cauchy sequence of ...
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45 views

Can $||f|| = ||a||_{q}$ to arbitrary values of $p$ and $q$ satisfying ${1 \over p} + {1 \over q} = 1$

We all know that: Suppose $a = (a_{1}, a_{2}, ..., a_{n})$ is a point in Euclide space $R^{n}$. Consider the mapping $f: R^{n} \rightarrow R$, $f(x) = \sum_{i=1}^{n}a_{i}x_{i}$. Then $||f|| = ...
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1answer
62 views

Hölder Space Definition

At the beginiing of the defintion of Hölder spaces and the Hölder space norm. They start defining the first term of the Hölder norm as follows: If $u:U \rightarrow \mathbb{R}$ is bounded and ...