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

learn more… | top users | synonyms

0
votes
1answer
31 views

How to prove a set is norm-closed?

I have to prove that the given space is 'norm-closed convex.' I proved the 'convex' part. But I don't know how to prove a set is 'norm-closed' I think I have to do the followings. Let X be a ...
1
vote
2answers
36 views

Showing existence of a vector $v$ such that $\|Av\|=\|v\|$ given two eigenvalues of the linear transformation $A$

Let $A : \mathbb R ^2 \rightarrow \mathbb R^ 2$ be a linear transformation with eigenvalues $\frac{2}{3}$ and $\frac{9}{5}$. Then, show that there exists a non-zero vector $v \in \mathbb R^ 2$ ...
4
votes
1answer
74 views

Fake proof: Equivalence of norms

Good morning. I'm having a hard time finding what's wrong with the following argument. Let $f$ be any function in $C^{1}([0;1])$ and let $||f||$ and $N(f)$ be two norms defined as follows: $$||f|| = ...
2
votes
1answer
29 views

Norm of Operator Proof

I'm stuck on this problem that I can't seem to figure out. Here's the problem. To note, equation 2.42 says that $$||T|| = \sup \{ ||Tu||: u \in C([a,b]), ||u|| = 1 \}$$ where $T$ is defined, ...
0
votes
1answer
13 views

Relation between a function and its norm

While reading up on Sturm-Liouville system theory, I came across something I didn't fully understand. At one point, in the midst of proving the existence of solutions to the Sturm-Liouvill problem, ...
1
vote
1answer
88 views

Linear and nonlinear operator on normed space and its properties

My first question is : We know every bounded linear operator is continuous operator , and every continuous linear operator is bounded operator , in other words the continuousness and boundedness are ...
0
votes
2answers
37 views

norms of Symmetric Positive Definite Submatrices

Let $\mathbf{A}\in\mathbb{R}^{N \times N}$ be symmetric positive definite. For some $1\leq k<N$, partition $$\mathbf{A}=\begin{pmatrix}\mathbf{A}_{11} & \mathbf{A}_{12} \\ \mathbf{A}_{12}^T ...
0
votes
1answer
41 views

Prob. 3, Sec. 2.8 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $C[-1,1]$ denote the normed space of all (real or complex-valued) functions defined and continuous on the closed interval $[-1,1]$ on the real line, with the norm given by $$\Vert x ...
5
votes
0answers
37 views

Matrix equivalence independent of dimension

I'm looking for a criterion on symmetric, positive definite matrices $A$ which ensures the constant $M$ in the lower bound of the norm equivalence $M\|A\|_\infty \le \|A\|_2 \le \|A\|_\infty$ does not ...
0
votes
1answer
36 views

What is the norm of a complex vector?

I have two arrays $a$ and $b$ containing complex values. Now I one of my target operations is the following: $$||a-b||$$ The result should be a single real number. Now I am a bit confused how to apply ...
3
votes
1answer
42 views

Is the trace of a matrix a norm?

If the matrix norm of A is defined as $\|A\|=\sum_{i,j}|Aij|$ then how do I determine if the sum of the diagonal elements, i.e., the trace is a valid norm? I am not really sure how to approach this ...
1
vote
1answer
41 views

$H_\infty$ 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: ...
0
votes
0answers
34 views

(Riesz's lemma) A closed subspace of a Banach space

Let V be a Banach space over R Let W be a proper closed subspace of V Prove : For any $\epsilon > 0$, there is a v $\in$_V_ such that ||v||=1 and ||v+w||$\geq$ $1$ - $\epsilon$ And my proof ...
1
vote
1answer
17 views

Parallelogram law using complex inner product not adding up

Does the parallelogram law still hold in the complex case? Using the following definitions: $\langle \textbf{x}, \textbf{y} + \textbf{z} \rangle = \langle \textbf{x}, \textbf{y} \rangle + \langle ...
1
vote
1answer
31 views

Banach space and its closed subspace. a vectors satisfying inequality.

V=a Banach space over R W=a proper closed subspace of V Prove : For any $\epsilon > 0$, there is a v $\in$_V_ such that ||v||=1 and ||v+w||$\geq$ $1$ - $\epsilon$ I have shown that there exists ...
1
vote
1answer
26 views

vector norms involving quantities with a filter relationship

I have two vector norm quantities: $\|\Psi^T(t)\Theta(t)\|$ and $\|\Phi^T(t)\Theta(t)\|$. Here $\Phi^T(t),\Psi^T(t)\in \mathbb{R}^{m\times n}$ and $\Theta(t)\in\mathbb{R}^{n}$. There is a filter ...
1
vote
1answer
22 views

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 ...
0
votes
1answer
48 views

Trace and Spectral norm of a matrix

Let $A_{n\times n}$ be a matrix. How I can show $$\vert \operatorname{trace} (A) \vert \leqslant n \sqrt{\rho(A^T A)}= n \Vert A \Vert_2$$ and if $A$ is symmetric and positive definite, ...
0
votes
1answer
46 views

How to write Holder's inequality for random vectors?

For $1 < p,q < \infty$ satisfying the constraint $1/p + 1/q =1$ and for $X, Y$ random variables such that $\mathbb E [\vert X \vert ^p ], \mathbb{E} [\vert X \vert ^q ] < \infty $ we have ...
0
votes
1answer
36 views

Operator matrix norm associated with the vector norm

What is the operator matrix norm associated with this vector norm? $$\Vert x \Vert = \frac{1}{n} \mathop {\sum} \limits_{j=1}^{n} \vert x_j \vert, \qquad (x \in \mathbb{R^n})$$
0
votes
1answer
47 views

Is it possible that $\Vert I \Vert > 1$ !?

For any matrix norm, is it possible $\Vert I \Vert > 1$ ?, where $I_{n\times n}$ is identity matrix. If not, why in some books they write $\Vert I \Vert \geqslant 1 $ ?
1
vote
0answers
40 views

$L^2$ vs $L^{\infty}$ norm for interpolation

Under what circumstances should I consider utilizing the $L^2$ norm instead of $L^{\infty}$ when interpolating a function based on sample points? Probably related: Does the answer significantly ...
1
vote
1answer
42 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 ...
0
votes
1answer
76 views

Prob. 14, Sec. 2.7 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $A \colon = [\alpha_{ij}]_{m\times n}$ be a given $m \times n$ matrix of real numbers. Let $\mathbb{R}^n$ be the normed space of all ordered $n$-tuples of real numbers with the norm defined as ...
4
votes
1answer
49 views

Compute $\| T \|$ with the norm $\| \cdot \| := \max_{j=1,\ldots,n} (|\cdot_j|)$

Let $A \colon = [\alpha_{ij}]_{m\times n}$ be a given $m \times n$ matrix of real numbers. Let $\mathbb{R}^n$ be the normed space of all ordered $n$-tuples of real numbers with the norm defined as ...
1
vote
1answer
44 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 ...
0
votes
1answer
28 views

is $N(f)=\int_{0}^{1} |f(t)|dt$ a norm on $E$(set of all continous real valued functions defined on [0,1])?

there are three conditions for a norm, I am stuck on one of them which is : $N(f)=0$ iff $f=0$. If i say $\int_{0}^{1} |f(t)|dt=0$ does this imply that $f=0$(zero function) ? Ok the other sense of ...
1
vote
2answers
64 views

Why does $(\vec{v}\cdot\vec{u})\vec{u}\neq(\vec{u}\cdot\vec{u})\vec{v}$?

I'm trying to save time running Gram-Schmidt. Why doesn't this product equal $||\vec{u}||\vec{v}$? More specifically (and I know this is fundamental and that I should already know it), why doesn't the ...
1
vote
1answer
57 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 ...
0
votes
2answers
43 views

Frobenius Norm to L2 norm Problem

Here is the problem: if $v^1$, $v^2$, ..., $v^d$ is an orthonormal basis in $\mathbb{R}^d$, then show that $$ ||A - A\sum_{i = 1}^k v^i(v^i)^T ||^2_F = \sum_{i = k+1}^d||Av^i ||_2^2. $$ I am having ...
4
votes
2answers
126 views

What is the norm of the pre-multiplication by a fixed matrix operator?

Let $A \colon= \left(\alpha_{ij} \right)_{m\times n}$ be a given $m \times n$ matrix of complex numbers, and let the operator $T \colon \mathbb{C}^n \to \mathbb{C}^m$ be defined by $$T(x) \colon= Ax ...
1
vote
0answers
36 views

Compute the value of the Sobolev norm in $H^{-1/2}$

I am working with finite elements using domain decomposition in 2D and one of the solutions I need to obtain is the co-normal derivative of the solution along a segment that is the intersection of two ...
0
votes
1answer
19 views

Infinity Norm calculation $\| \ln(x) - (\ln(3/2) + 2/3(x-3/2)) \|$

I have the following infinity norm: $$ \| \ln(x) - (\ln(3/2) + 2/3(x-3/2)) \|_\infty. $$ Computing from [1,2]. I know that I can compute this in matlab and I get .072. However, how would one go ...
0
votes
0answers
28 views

Differentiation of second norm

Considering $f = {e^{ - {{(||x - y|| - R)}^2}}}$ where ||.|| denotes the second norm, and $x \in {R^n}$. The following derivative is correct? $\frac{{\partial f}}{{\partial x}} = - 2(||x - y|| - ...
2
votes
1answer
24 views

Norm of orthonormal basis

I know that an orthonormal basis of a vector space, say V is a orthogonal basis in which each entry has unit length. My question is, then, if you have some orthonormal basis say $\{v_1,\ldots,v_8\}$ ...
1
vote
1answer
28 views

When does the $L_1$ convergence imply almost everywhere convergence?

I know that $L_1$ convergence implies existence of an almost everywhere converging subsequence. But I was wondering, can you tell me some extra conditions on functions that make $L_1$ convergence ...
1
vote
1answer
41 views

Supremum vs. Maximum in the definition of the Lp norm [duplicate]

The $L_p$ norm $||A||_p$ is defined as $$\sup_{x \neq 0} \frac{||Ax||_p}{||x||_p} = \max_{||x||_p = 1} ||Ax||_p \tag{1}$$ I'm not quite getting why the LHS uses $\sup$ but the RHS uses $\max$. I ...
0
votes
0answers
28 views

How would I calculate the error in LU Decomposition of a matrix?

I am currently practicing the LU Decomposition of n x n matrices. After finding the LU decomposition, I am lost on how to find the error. I am trying to understand the notation in a problem that asks ...
2
votes
0answers
36 views

Proving inequality that bounds the sum of norms with the norms of sums (plus additional terms)

I am struggling with showing the following for finite $\delta>0$ and any $g\in\mathcal{G}_1\times...\times\mathcal{G}_k$: ...
0
votes
0answers
30 views

Why is this an nuclear norm?

The following figure is from a paper: "The Convex Geometry of Linear Inverse Problems" Just on p.3. As in the figure, the red lines are $2 \times 2$ symmetric unit-Euclidean-norm rank-one ...
0
votes
0answers
17 views

The meaning of equivalence in norm.

Ask a elementary question: In WIKI: " However all these norms are equivalent in the sense that they all define the same topology." I think "these norms" here mean $l_1, l_2,...l_{inf}$ norms. ...
0
votes
0answers
27 views

On norms for “more complicated objects”

Whilst studying norms (via various routes) I was always confused by the following definition for some object $\phi$, $$\|\phi\|_p=\sup_{x\in A,x\neq 0}\frac{\|\phi x\|_p}{\|x\|_p}=\sup_{x\in ...
1
vote
1answer
21 views

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 ...
5
votes
2answers
42 views

Proving an inequality involving norms of real functions.

If $ f : [a,b] \subset \mathbb{R} \rightarrow \mathbb{R} $ is continuous and differentiable in $(a,b)$, then one can define a norm for such functions as $$ \|f\| = |f(a)| + \max_{x \in (a,b)} |f ...
0
votes
1answer
36 views

Functional analysis, normed space

Can you please help me with this exercise.. I have to check if in interval $[a,b]$ continuously differentiable functions $x=x(t)$ norm can be defined as: $$\vert x(b) - x(a) \vert + \max_{a \leq t ...
4
votes
3answers
90 views

Can one find a stronger norm on a Banach space?

Given a Banach space $V$ of infinite dimension with norm $\|\cdot\|_1$, is that possible to find a norm $\|\cdot\|_2$ on $V$ such that the topology induced by $\|\cdot\|_2$ is strictly stronger than ...
0
votes
1answer
56 views

Proof of relation between maximum element and induced $p$-norm of a matrix

If true, prove the identity: $$ ||A|| \ge \max\limits_{i,j}|a_{ij}| $$ $||.||$ is any induced/operator norm. Edit: The identity is true only for operator norm induced by $p$-norm for vectors. I ...
1
vote
1answer
32 views

Derivative of a Radial Basis Function $\nabla_x \frac{1}{1+\|x - y\|^d}$

I want to calculate the following derivative (with respect to x): $\nabla_x \frac{1}{1+\|x - y\|_2^d}$ where, $x,y \in R^D$ and $d$ is some positive integer. $\| \|_2$ is a 2 norm of a vector.
1
vote
2answers
32 views

How can I prove this operator is not continuos

Let $X$ the normed space of all polynomials on $J=[0,1]$ such that $||x||$=max$|x(t)|$ $t \in[0,1]$ and we have the following operator $Tx(t)=x'(t)$ prove this operator is not continuous
0
votes
0answers
10 views

Relation between max element norm and other norms of a matrix

I am looking for a relation between max element norm and other norms like $2$-norm or $\infty$-norm (in terms of bounds) so that I can reduce the following condition $$ ...