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

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$H_\infty$ norm of a transfer function by Matlab

Consider the simplest case: $$H(z) = \frac{z+1}{z+2}$$ I use two methods to find $H_\infty$-norm: ...
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Existence of a vector whose norm is 1 in a Banach space

Given a finite dimensional Banach space V over reals, I have to show that there exists $v \in V$ such that $\|v\|=1$ At first, I thought that there's an identity element I in V. And $\|I\|=1$. But ...
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50 views

Why is $A:X\to Y$ linear between two normed spaces is continuous iff bounded?

Why is it that every linear operator $f: \mathbb R^n \to \mathbb R^m$ is bounded and therefore continuous, but why is it that $A:X \to Y$ between two normed spaces is continuous iff bounded? That ...
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64 views

Upper bound for norm of matrix (cf. Example 2.7-7 in Erwine Kreyszig's book)

Let $A \colon = [\alpha_{ij}]_{m\times n}$ be a given $m \times n$ matrix of real numbers. Let $\mathbb{R}^n$ be the norm space of all ordered $n$-tuples of real numbers with the norm defined as ...
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114 views

Holder's inequality with expectation norm

One defines the ``p-expectation norm" of a function $f$ as $\vert f \vert_p = (\mathbb E ( f^p) )^{ \frac{1}{p}}$. What is the intuition for this norm? Now why are the following true? $\vert ...
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Norm of a difference of two elements in Quadratic Fields

everyone. Hope you can help me out. I ran across this question and have absolutely no idea how to go about: "Let $X =x+y\sqrt{2}$ with $x,y\in\mathbb{Q}$. Show that there is an element ...
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47 views

I can't understand one step of this matrix norm proof

I'm reading a multivariable calculus textbook for college, and before a Taylor series proof is given, a lemma is provided with its demonstration. The lemma says: Given a matrix $M(X)= ...
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22 views

Let $u,v \in \mathbb{R}^n, A = u\cdot v^T$ then prove $\|A\|_2 = \|u\|_2 \; \|v\|_2$

Let $u,v \in \mathbb{R}^n, A = u\cdot v^T$ then prove $\|A\|_2 = \|u\|_2 \; \|v\|_2$ How can I do this? I'm kinda stuck with this 2-norm of the matrix. If it would be $\|A\|_{\text{frob}}$ then ...
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51 views

Bounded functions, existence of a functional with a given norm

Let $\mathcal{B} ([0,1])$ be the space of all bounded real functions on $[0,1]$ with sup norm. Prove that for each $x_0 \in [0,1]$ there exists a functional $\varphi \in \mathcal{B} ([0,1])'$, $|| ...
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82 views

Convergence in norm and in mean

Here is my problem: If $\{f_k\}$ is a sequence in $L^2$ and $f_k\to f$ in mean, show that $\{||f_k||_2\}$ is a bounded sequence of real numbers. Before I start doing the problem, I would like to know ...
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32 views

Equality between 2-Norm of polynomials

Let $f(x)=a_nx^n+a_{n-1}x^{n-1}+ \dots a_0$ be a polynomial of $\mathbb{C}[x]$, we define $$||f||_2= \sqrt{ \sum_i|a_i|^2}$$ Now, studying Computational Algebra, I've read about the following ...
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60 views

How to “compare” vectors?

I'm reading the definition of matrix norm in Golub & Van Loan and came across this "It is clear that the p-norm of matrix A is the p-norm of the largest vector obtained by applying A to a unit ...
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84 views

Polynomial, bounded functional

In order to prove continuity of the functional $$\varphi: \mathbb{R}[X] \ni p \rightarrow p'(2011) \in \mathbb{R}$$ where $$||p|| = \sup \{|p(t), \ t\in [0,1]\}$$ I'd like to prove that this ...
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66 views

Which is finer(larger) between the sequence spaces $l_{p}$ & $l_{p+1}$

Prove that, $l_{3}\subset l_{7}$ & $L_{9}[0,1]\subset L_{6}[0,1]$, where $l_{p}$ & $L_{p}[0,1]$ are of usual notation. Are the converses hold for both cases? Can these two results ...
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56 views

Polynomials of best approximation

The question is about approximating the continuous function in an interval $[a, b]$. If we consider the linear space of all such functions endowed with the norm $$||f|| = \max_{x \in [a, b]}|f(x)|$$ ...
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104 views

What is the general definition of the conjugate of a multiple-component number?

I know that the conjugate of a binomial is the negation of the second part. So the conjugate of (a + b) would be (a - b). I know that the conjugate of a complex number (a + bi), similarly, is (a - ...
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58 views

$\|u\| = \Big(\int |\nabla u|^p \Big)^\frac{1}{p}$ as a norm on $W_0^{1,p}(\Omega)$ where $\Omega$ is bounded

Let $\Omega$ be a bounded open set of $\mathbb{R}^N$, define $\|u\|_1 = \Big(\int |\nabla u|^p \Big)^\frac{1}{p}$ as a norm on $W_0^{1,p}(\Omega)$ for $2\leq p<\infty$, where $$\int |\nabla u|^p ...
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47 views

Limits in matrix norm if convergence of integration is guaranteed

the answer of this question probably is very obvious, but I want to make sure this is correct. I have a function $F: \mathbb{R}\rightarrow \mathbb{R}^{n\times n}$ that is continuous and ...
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41 views

Understanding an equality with open balls

In order to understand a proof I want to know why the following is true: Let X be a banach space and $x\in X$ with $x\in B(x_0,r)$ (open ball around $x_0$ with radius $r$), then $\frac{r}{2}x-x_0 \in ...
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Is $\bigl\|\frac{vv^T}{v^Tv}\bigr\|=1$? For any vector $v\in \mathbb{R}^{n}$

I am stuck while showing that $$\biggl\|\frac{vv^T}{v^Tv}\biggr\|=1,$$ where $v\in \mathbb{R}^n$, and $\|.\|$ is a matrix norm. Here is my steps: I used Frobenius norm: A Frobenius matrix ...
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79 views

Convergence as for the norm [duplicate]

If $f_n, f \in L^p, 1\leq p < +\infty$ and $f_n \rightarrow f$ almost everywhere, and $\|f_n\|_p \rightarrow \|f\|_p$, then $f_n\rightarrow f$ as for the norm. Could you give me some hints how to ...
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40 views

Does this funcion define a norm on $\mathbb{C}^n$?

Let $m$ and $n$ be two given positive integers. And, let $f \colon \mathbb{C}^n \to \mathbb{R}$ be defined as follows: $$ f(x_1, x_2, \ldots, x_n) \colon= \left( \sum_{i=1}^n \sqrt[m]{|x_i|} ...
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48 views

Prove the norm inequality.

Exercise. Let $p_1$, $p_2$ be norms on $\Bbb R^n$ with respective unit balls $D_!, D_2$. Prove that $$D_2\subset D_1\iff p_1(x)\le p_2(x)\ \forall x\in\Bbb R^n$$ Can someone please help me this. ...
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23 views

When is $\| X \| _\star = \| F X \| _2$ submultiplicative?

All matrices are real. By $\| \cdot \|_2$ denote the matrix norm induced by $L_2$. Assume $F$ is an invertible matrix. Consider the norm $\| X \| _\star = \| F X \| _2$. What is the condition on ...
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140 views

Finding the norm of integral operator

I have the following operator: $A: (C^1[0;1];|||\cdot|||)\rightarrow(C^1[0;1];||\cdot||_{\infty})$ $Af(x)=\int_0^x f(t)dt$ where $|||f|||= ||f||_\infty+||Af||_\infty$ What is $||A||?$ I wrote ...
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99 views

Can you divide by matrix norms in an equation?

Supposing that a matrix A has an eigenvalue lambda, show that for any induced matrix norm, $||A|| \geq |\lambda|$. I attempted the solution, but I am not sure if it is valid to cancel the norm of ...
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36 views

Strengthening bound in the Euclidean space

Suppose that $z$ is a $K\times 1$ vector. Denote the components of $z$ as $z_1,\ldots, z_K$. Let $r>0$ be given. I'd like to find the smallest constant $C$ such that $$ |z|^r\leq ...
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228 views

Condition number - proof

I have a problem with which I've been struggling for a while. It's probably not that difficult, but I seem to be stuck so... here we are. Let A be an invertible lower- or upper-triangular matrix ...
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223 views

Norm preserving Matrix properties

Norm-2 preserving can be done using unitary/orthogonal matrix: $A^*A = I => ||Ax|| = ||x||$ What is the matrix other than identity matrix that can preserve other norms ( norm-1, norm-inf) ?
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87 views

Dual Optimization Problem

I have the following optimization problem, $$ \text{minimize}_{X,Y} \ \lVert X\rVert_* + \lambda \lVert Y\rVert_1 \\ \text{subject to }X + Y = C$$ $C \in \mathcal{R}^{m \times n}$, $\lVert ...
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63 views

How to check if a given piecewise defined function on $\mathbb R^2$ is a norm?

I want to check if the function $\parallel (x,y)\parallel = \left\{ \begin{array}{cc} \sqrt{x^2+y^2} & \mbox{if } xy \geq 0 \\ \max\{\vert x\vert, \vert y\vert\} & \mbox{if } xy < 0 ...
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63 views

Show there exists a Cauchy subsequence

Let $X$ be a separable reflexive real Banach space and $\{\psi_n\}$ be a dense sequence in $$\{\psi\in X' : ||\psi||_{X'} \leq 1\}.$$ Consider in $X$ the scalar product defined by $$(x | y)_0 = ...
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Well-Posedness PDE of the Form $\partial_t u = P(\partial_x) u$ for a Polynomial $P$

My question is to determine whether the PDE $\partial_t u = P(\partial_x) u$, with $2\pi$-periodic boundary conditions, for a polynomial $P$, is well-posed; this depends on the polynomial, and my ...
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32 views

Equivalence of sets

Let $u_1, u_2, u_3 \in \mathbb{C}$ be the cubic roots of unity I'm wondering if the following two sets (balls) are equivalent: $$ \lbrace (v,w) \in \Bbb C^2 : \vert v \vert + \vert w \vert \leq 1 ...
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591 views

Why is L21 norm not smooth

I have this confusion. I was reading this paper http://www.cis.temple.edu/~yuhong/research/papers/ijcai13b.pdf. I didn't understand why is L21 norm not smooth?
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42 views

Prove or disprove the existence of a length preserving non-normal matrix

Prove or disprove: There exists A such that $\|Av\|=\|v\|$ for any $v \in R^n$, but $A$ is not a normal matrix There exists A such that $\|Av\|=\|v\|$ for any $v \in R^n$, but $A$ is not a unitary ...
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52 views

Show that a set is compact.

Let $X$ be a Banach space and $\{A_t\}_{t\in R}$ a family of linear and continuous maps $X \rightarrow X$ such that function $\mathbb{R} \ni t\rightarrow \|A_t x\| \in \mathbb{R}$ is continuous for ...
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62 views

Find the norm of functional

Consider the functional from $l_2$. $$ x=(x_n)\mapsto \sum \frac{x_n+x_{n+1}}{2^n}. $$ What is the norm of the functional?
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Show that $\int_{\mathbb R^n}e^{|x|^{-n}}dx=$ Volume of n-sphere

I'm preparing for a calculus exam, I'd like help in solving this question. Let $x \in \mathbb R^n$, $|x|={(x_1^2+x_2^2+...+x_n^2)^{\frac{1}{n}}}$, Show that $$\int_{\mathbb R^n} e^{|x|^{-n}}dx$$ is ...
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229 views

Relationship between matrix 2-norm and orthogonal basis of eigenvectors

Given the following matrix: $$ A = \left( \begin{array}{cc} 3 & 4 \\ 0 & 5 \\ \end{array} \right)$$ calculate $\|A\|_2$, with $\|A\|_2 = max_{x \in \mathbb{R}^2 -\{0\}} \frac{\langle Ax,Ax ...
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82 views

Inequality with a norm

I need help with the following: Let $A=\left(\begin{array}{cc}a & b \\c & d\end{array}\right)$, with $a\in\mathbb{R}$, $b\in(l^{1})^{*}$, $c\in l^{1}$, and $d\in L(l^{1},l^{1})$. Let $h\in ...
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226 views

Why is there an “absolute value” and a norm in the Schwarz Inequality?

This really bothers me, and I'm not sure if it's just that I'm not understanding it correctly. For the moment, assume we are working in a vector space $V$ over $\mathbb{R}^n$. Let $x,y \in V$. We have ...
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270 views

Fredholm operator norm

I have seen here, that the operator norm of a Fredholm operator $T_k(f)(s):=\int_0^1 k(s,t) f(t) dt $, where $k \in L^2([0,1]^2)$ and $f \in L^2([0,1])$ is not equal to the $L^2$ norm of the Kernel. ...
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Prove $||A||_2 = max_{x \neq 0, y \neq 0}\{\frac{|\langle y,Ax \rangle|}{||x||\cdot ||y||}\}$ for $A\in \mathbb{K}^{n\times m}$

I'd like to prove that the spectral norm of a matrix that is not necessarily square can be written as the following subordinate norm $||A||_2 = max\{\frac{|\langle y,Ax \rangle|}{||x||\cdot ||y||}, y ...
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81 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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43 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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82 views

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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58 views

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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62 views

Prove scalar product in a normed vector space is an open mapping.

I have the feeling this is a really obvious question, but I'm having trouble with it, here it goes: Let $(X, ||\,||)$ be a normed vector space over $K$, prove that $\odot:K\setminus\{0\}\times X ...
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54 views

Norm of linear functional

For every $ x \in C([a,b])$, we define the functional $F(x) = \sum_{i=1}^{n} {\lambda_{i} x(t_{i})}$ where $\lambda_{i} \in R, i=1,...,n$. I was wondering if someone can help me to find a sequence ...