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
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 ...
1
vote
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 ...
1
vote
1answer
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|| = ...
1
vote
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 ...
1
vote
1answer
230 views

Condition Number of a block Matrix

Is this hypothesis true? $$cond([A,B])≤cond(A)+cond(B)$$ where $cond$ is the Condition Number. And is this true for rectangular matrices($nxm$)? Let's consider $3$ different conditions for $A$ and ...
0
votes
1answer
47 views

Generalized minimal residuals: eigenvalues and sets of functions

Can someone help me on this exercise (2 parts)? Thanks! Suppose that $S \subseteq \mathbb{C}$ is a set whose convex hull contains $0$ in it's interior (so $S$ is contained in no half-plane ...
1
vote
1answer
94 views

Equality involving norm on Cholesky/QR factorization

Let $B$ be symmetric and positive definite and $B=B^{\frac{1}{2}}B^{\frac{1}{2}}$ the Cholesky factorization. Having $A=QR$, why can we follow the last equality in the following? $$ || A ...
0
votes
2answers
233 views

inequality with the Frobenius norm for matrices

Let $A\in M_n$. How can I show that $$\left|{\textrm{Tr}(A)\over\sqrt{n}}\right|\leq \Vert A\Vert_F$$ I tried it using the Cauchy-Schwarz inequality.
2
votes
2answers
350 views

Vector norm Inequality proof

Does anyone know how to start proving this inequality $$ \left\|\frac{x}{\|x\|} - \frac{y}{\|y\|}\right\| \leq \frac{4 \|x-y\|}{\|x\|+ \|y\|} $$ The norm is a random norm on a vector space $V$
2
votes
1answer
64 views

$\lambda_k \to 0$ 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 ...
0
votes
1answer
37 views

Methods of computing the derivative of vector norms

I am very new to norms. Except the basic definitions and properties of the norm, I don't know too much about it. Now, I am very interested in computing the derivative of the norms. So, I am wondering ...
2
votes
2answers
162 views

Proof that two norms $\lVert\cdot\rVert_1$ and $\lVert\cdot\rVert_2$ are equivalent

Two norms $\def\norm#1{\lVert#1\rVert}\norm\cdot_1$ and $\norm\cdot_2$ are equivalent iff $\;\exists\;c_1,c_2>0$ such that $c_1\norm x_1\le \norm x_2\le c_2\norm x_1$ Show that $\norm ...
2
votes
1answer
126 views

continuity of norms with respect to $p$

Let $f\in L^{\infty}(\Omega,\Sigma,\mu)\cap L^{1}(\Omega,\Sigma,\mu)$. Then $w(p)=||f||_p$ is continuous function of $p$ for any $p\in [1,\infty)$. How to prove this? I have obtained the proof that ...
3
votes
2answers
189 views

Norm of random vector plus constant

Suppose that $w$ is a multivariate standard normal vector and $c$ a real vector of the same size. I know that for positive x $$P(||w+c||^2\geq x)\ \geq \ P(||w||^2\geq x)$$ but I cannot prove it. We ...
0
votes
1answer
50 views

Does this show the norm of this operator is zero?

We have $$T: C[-1,1]:\to \mathbb{R}$$ $$T(f)=\int_{-1}^1 x f(x) dx$$ The norm considered in $C[-1,1]$ is $$||f||=\max_{x\in[-1,1]} |f(x)|$$ So using $$||T||=\inf\{M:||Tf||\leq M||f||\}$$ in this ...
1
vote
1answer
154 views

does the max function holds the triangle inequality?

I need to prove if the following is a norm: $$||f||:=\max_{-1<x<0}|f| + \max_{0<x<1}|f|$$ when $f$ is a continuous on $[-1,1]$. The only problem I have is with showing it holds the ...
1
vote
1answer
117 views

Condition Number of a Product

Is this hypothesis true? $$\operatorname{cond}(AB) \leq \operatorname{cond}(A)\operatorname{cond}(B),$$ where $\operatorname {cond}$ is the condition number. And is this true for rectangular matrices? ...
2
votes
0answers
162 views

Using the Sobolev-Nirenberg-Gagliardo inequality in a proof

If $1 \leq p < n$. The Gagliardo-Nirenberg-Sobolev inequality states that there exists a constant C such that $||u||_{L^{p^{*}}(\mathbb{R}^{n})} \leq C||Du||_{L^{p}(\mathbb{R}^{n})}$ for all $u ...
1
vote
0answers
107 views

Triangle inequality for $\sqrt{x^T\cdot A\cdot x}$

I want to prove that $f(x):= \sqrt{x^TAx}$ is a norm, $A \in \mathbb{R}^{n \times n}$ positive definite, $x \in \mathbb{R}^n$. I already proved its positivity and absolute homogeneity, but I don't ...
0
votes
2answers
96 views

Limit of nth power of operator norm

I am given a compact operator $A$ which lives in a Banach algebra and whose spectral radius obeys $\rho(A)=\lim_{n\rightarrow\infty}||A^n||^{\frac{1}{n}}<1$. Now I want to prove that this implies ...
0
votes
1answer
38 views

Given identity map $U:\ell^n_a\rightarrow \ell^n_b$ for $\mathbb{R}^n$, how to computer operator norm $\forall a,b$?

If $U:\ell^n_a\rightarrow \ell^n_b$ is the identity map of the underlying vector space $\mathbb{R}^n$, then how do you compute the operator norm $U$ for all possible values of $a$ and $b$?
1
vote
1answer
53 views

Hints on calculating an integral

Suppose $x(t)$ and $y(t')$ are curves traversing the boundary of $[0,1]^2$ in $R^2$ counterclockwise. What is the integral of the following: $$\int\limits_{t,t'}{\frac{dt\,dt'}{\|x(t)-y(t')\|}}$$ I ...
0
votes
1answer
38 views

Showing that function are equal almost everywhere in Sobolev Spaces

Consider the Holder space $C^{0,1-\frac{n}{p}}(\mathbb{R}^{n})$ and the Sobolev Space $W^{1,p}(\mathbb{R}^{n})$. Take $u_{m} \in C_{c}^{\infty}(\mathbb{R}^{n})$ such that Morrey's Inequality we have ...
1
vote
1answer
70 views

Space on convergent sequences - is it an inner product space

I'm trying to prove some properties of sequence spaces. I already know that the space $l^{\infty}$ of all bounded sequences isn't an inner product space, isn't separable but it is complete with $sup$ ...
0
votes
1answer
39 views

If a matrix of the form I + B is singular, then ||B|| ≥ 1 for every subordinate norm.

I need some guidance showing that: If a matrix of the form I+B is singular, where I is the identity matrix, then for any subordinate norm $\|\cdot\|$, $\|B\|\geq1$.
0
votes
0answers
24 views

Norm of the maximum

Consider the norm $||f||= max_{x\in[a,b]} |f(x)|$ defined in the bectorial space $C[a,b]$ I have to what is the meaning (/interpretation) in $R$ of {$||f_n-f||$}$\to$ 0 Could you help me?
1
vote
2answers
213 views

A norm for Lipschitz-continuous functions

I am a first year undergrad math student, and I am struggling with a proof. Let the set $C_{\text{Lip}}:= \left \{ f:\mathbb{R}\rightarrow \mathbb{R}: f \text{ is Lipschitz continuous} \right \}$ be a ...
2
votes
1answer
50 views

Minimization of norms

How do I minimize the following? $ min_{z>0} - zt + 1/2\ z\ ||\ Y + X_k\ /\ z\ ||_2^2 $ Also, $X_k^TX_k = 1 \ \ \forall k $ I am given that the answer should be : $ \sqrt{Y^T - 2t} + Y^TX$ ...
0
votes
1answer
34 views

Square of 2-norm

This might be silly but I am stuck with the following problem: $ || Y - Z_i/x||^2_2 $ = 2t How would I solve to get $x $ from this equation?
2
votes
1answer
34 views

What could be the lead to prove $||X||_2 \leq ||X||_F \leq \sqrt{rank(X)}||X||_2$?

In the above statement, $||X||_2$ = $L_2$ norm of X and $||X||_F$ = $Frobenius$ norm of X. It appears to me that the $L2$ norm of X and $Frobenius$ norm of X are the same. How should i proceed to ...
3
votes
0answers
138 views

Two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ on $X$, and a function $f:X\to Y$ Fréchet differentiable with one of the norms but not with the other one?

There is a theorem that if $f : (X,\|\cdot\|_{X1}) \to (Y,\|\cdot\|_{Y1}) $ is Fréchet differentiable, then replacing the norms with some equivalent norms $\|\cdot\|_{X2}$ and $\|\cdot\|_{Y2}$ ...
5
votes
1answer
379 views

Dual to the dual norm is the original norm (?)

I have the following questions about dual norms : How do you prove that the dual of the dual norm is in fact the original norm? This is what I have so far: If I have $\|y\|_* $ as the norm dual of ...
0
votes
1answer
157 views

Low-rank matrix approximation in terms of entry-wise $L_1$ norm

According to the Eckart–Young theorem, the low-rank matrix approximation problem $$\min_{\tilde{A}} \quad \| A - \tilde{A} \|_F \quad \text{s.t.} \quad \text{rank}(\tilde{A}) \le r$$ is given by the ...
1
vote
1answer
61 views

Example of two norms and ONE linear operator that is bounded and unbounded in a norm.

I am looking for an example of a linear operator that is bounded as well as unbounded depending on which norm you take. Since I do not have much experience with Functional Analysis, I do not know many ...
0
votes
2answers
94 views

Norm space, linear operator exercise, help please!

$f \in L_2[a,b]$ $Uf(s):=\int_a^bk(s,t)f(t)dt$ $k(s,t):[a,b]^2\to R $ continuous. show 1) $U:L_2[a,b]\to L_2[a,b]$, in other words, $Uf(s)\in L_2[a,b] \quad \forall f$ 2) $U$ is linear and ...
3
votes
2answers
167 views

Poincare Inequality implies Equivalent Norms

I am currently working through the subject of Sobolev Spaces using the book 'Partial Differential Equations' by Lawrence Evans. After the result proving the Poincare Inequality it says the following ...
0
votes
1answer
46 views

Help in proving that a vector norm satisfies an axiom.

I am trying to prove if the following is a vector norm: ||x|| = max{$|x_1 + x_2|, |x_2 + x_3|, |x_3 + x_1$|} (x is vector with 3 elements) I'm stuck proving that $||\alpha x||=|\alpha|*||x||$. I ...
2
votes
0answers
41 views

whether the function p(x)=||x|| + |||x||| defines a norm where ||.|| and |||.||| are two different norms [closed]

whether the function p(x)=||x|| + |||x||| defines a norm or not, where ||.|| and |||.||| are two different norms. i have a clue to check for the axioms of the norms. but tel me how to start?
0
votes
1answer
50 views

Calculate norm $1$ of $f(x)=2x^3+3x^5$ belonging to C[-1,1]

Calculate norm $1$ of $f(x)=2x^3+3x^5$ that belongs to $C[-1,1]$. As norm $1$ is called integral norm, I calculated the value of the function for the given interval, and the answer I get is zero. ...
-3
votes
1answer
70 views

Conditional number: exercise

Let's say we've got a $202 \times 202$ matrix $A$ for which $||A||_2=100$ and $||A||_F=101$ (the Frobenius norm). How can we find the sharpest bound (lower) on the 2-norm condition number of $A$? ...
2
votes
0answers
62 views

Asymptotic behavior of $L^2$ norm for increased matrix dimensions

I am playing with matrices which are linear combinations of identity matrix, Pauli spin matrices, $\sigma_x$ and $\sigma_z$ or their tensor products. For example, let the matrix be $H$. So, $H$ could ...
0
votes
2answers
92 views

Minimize integral

Find numbers A and B such that the integral is minimal $$ \int_{0}^{\infty}\left\vert% \,\vphantom{\Large A}{\rm e}^{-x} - A{\rm e}^{-2x} - B{\rm e}^{-3x}\, \right\vert^{2}\,{\rm d}x $$ I have tried ...
4
votes
1answer
56 views

Is it true that $2^p\|f\|^p_{L_p}+2^p\|f'\|^p_{L_p} >2\|f\|^p_{L_\infty}$?

For any $C^1$ function defined in $(0,1)$, is it true that $$ 2^p\|f\|^p_{L_p}+2^p\|f'\|^p_{L_p} >2\|f\|^p_{L_\infty} $$ If it is true, how to prove it?
0
votes
1answer
43 views

Is it true that $2^p\|f\|^p_{L_p} >2\|f\|^p_{L_\infty}$?

For any $C^1$ function defined in $(0,1)$, is it true that $$ 2^p\|f\|^p_{L_p} >2\|f\|^p_{L_\infty} $$ If it is true, how to prove it?
0
votes
2answers
93 views

Is $L_\infty$ norm the smallest or largest?

I am a little bit confused. For a $L_p$ function norm, is it true that for any $ p<\infty $, $$ \|f\|_p>\|f\|_\infty$$ Is the statement true for any domain? I want to know more inequality about ...
0
votes
1answer
183 views

Finding a Unit Vector v for a Matrix A such that the 2-norm of AV is equal to the 2-norm of A

I have been working on the following problem: Let A be the following 2x2 matrix: A = [1 1; 0 1] (MATLAB notation) Find the 2-norm of A and a unit vector v such that the 2-norm of Av = the 2-norm of ...
0
votes
2answers
35 views

Useful relationships that are true for every norm

I am looking for useful identities that are true for every normed vector space $(V,||.||)$ on either $\mathbb{R}$ or $\mathbb{C}$(if your identity is restricted to either one of them, please say so). ...
3
votes
1answer
50 views

what is the limit of $l_p$ at p=0?

The p-norm is defined as: $$ \ \|x\|_p=\left(|x_1|^p+|x_2|^p+\dotsb+|x_n|^p\right)^{\frac{1}{p}} $$ When $p<1$, this is no longer a "norm" because it violates the triangle inequality (- it is ...
1
vote
2answers
65 views

Vector norms on $\Bbb R^n$

Let $u=(u_1,u_2,\ldots,u_n)$ and $v=(v_1,v_2,\ldots,v_n)$ be two vectors in $\Bbb R^n$. Suppose $\left|v_i\right|>|u_i|$ for all $i$. Let $\| \cdot\|$ be any vector norm on $\Bbb R^n$. Is it true ...
3
votes
3answers
106 views

evaluating norm of sum of roots of unity

let $l_1,...,l_n$ be roots of unity. I want to prove that the norm(the product of all conjugates)of $a=l_1+...+l_n$ is not greater than $n$, not smaller than $-n$. how can I do to prove this?