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

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457 views

Show these two norms are not equivalent?

I have the following two norms on $C[a,b]$ : $$||x||_1= \int_a^b |x(t)|dt$$ $$||x||_\infty = sup_{t \in [a,b]}|x(t)|$$ $\forall x \in C[a,b]$. I need to prove that these are not equivalent. They are ...
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1answer
44 views

Prove this sequence in $L(\ell_1)$ does not converge to zero?

I have a linear operator $$A: \ell_1 \rightarrow \ell_1$$ $$A_nx=(0,0,...,0,x_{n+1},x_{n+2},...)$$ with $n$ zeros. I am asked to show two things: that for any $x \in \ell_1$, $\lim_{n \rightarrow ...
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2answers
70 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 ...
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1answer
50 views

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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0answers
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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1answer
177 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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1answer
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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1answer
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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1answer
61 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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1answer
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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0answers
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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1answer
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. ...
3
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1answer
284 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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1answer
474 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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1answer
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 ...
2
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1answer
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} ...
2
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1answer
90 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 $K$ ...
4
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1answer
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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1answer
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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1answer
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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1answer
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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2answers
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. ...
2
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1answer
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 ...
2
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1answer
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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1answer
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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2answers
48 views

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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1answer
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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1answer
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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4answers
469 views

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 ...
0
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1answer
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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2answers
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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1answer
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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2answers
47 views

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 ...
2
votes
1answer
49 views

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 ...
0
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1answer
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 ...
1
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0answers
53 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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1answer
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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1answer
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 $1$ ...
2
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1answer
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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1answer
28 views

$|Av||A^{-1}v|$, $A$ 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 ...
0
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1answer
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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1answer
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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1answer
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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2answers
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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1answer
420 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 ...
0
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1answer
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 ...
1
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2answers
90 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 ...
0
votes
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 ...
0
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1answer
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 ...