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

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76 views

A norm on $\mathbb{R}^2$ such that $\partial C$ is the unit sphere?

Suppose we are on $\mathbb{R}^2$. Assume that $C \subset \mathbb{R}^2$ is a convex bounded neighborhood of the origin invariant by central symmetry. Let $\partial C$ denote the boundary of $C$. My ...
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2answers
55 views

Condition for a norm be absolute

Let $\|x\|_B\mathrel{\mathop:}=\sqrt{x^{t}Bx}$, where $B \in \mathbb{R}^{n\times n}$ is a symmetric and positive semidefinite matrix. If $\mid x\mid = (|x_1|,|x_2|,\ldots,|x_n|)$, I want to show that ...
1
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1answer
60 views

$||f + g||_{L^p} = ||f ||_{L^p} +||g||_{L^p} $ for $p\in\mathbb{R}^+\setminus\{ 1\}$ and $f,g\geq 0$

$||f + g||_{L^p} = ||f ||_{L^p} +||g||_{L^p} $ for $p\in\mathbb{R}^+\setminus\{ 1\}$ and $f,g\geq 0$, then $f = Cg$ for some non-negative constant $C$. First assume $||f ||_{L^p} +||g||_{L^p} = 1$, ...
2
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1answer
59 views

Proof that frobenius norm is a norm [duplicate]

It's pretty basic and I'm sure I'm missing something dumb here, but I'd like to know why $||A+B||_F \leq ||A||_F+||B||_F$ The way I understand it, ...
0
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1answer
30 views

2-Norm of a complex matrix equation

I am having trouble understanding the following excerpt from a math text I'm working through: My question specifically is how line 2 came about in the expansion. How do the real and imaginary parts ...
2
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1answer
95 views

Is the matrix least squares minimizer (Frobenius norm) the same as the matrix 2-norm minimizer?

Given matrices $A \in \mathbb{R}^{n \times m}$ and $B \in \mathbb{R}^{n \times k}$, consider the (least squares) minimizer $\arg \min_{X \in \mathbb{R}^{m \times k}} \| AX - B \|_F$, where $\| M \|_F$ ...
0
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1answer
29 views

Cauchy sequence such that don't have limes in C[0,1]

Give an example of series $f_n \in C[0,1]$ such that $f_n$ is Cauchy sequence in norm $$\|(a_n)\|_p = \left( \sum_{n=1}^{\infty} |a_n|^p \right)^{1/p}$$ and $$\lim_{n \to \infty} f_n(x)$$ don't ...
0
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1answer
24 views

help about find the norm in$L^1$ [duplicate]

I need to find norm of $f$.Firstly I try to show boundedness and so I need to find any $M>0$.But coefficient of integral is zero.Where is my mistake? and I used holder inequality for norm in ...
2
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1answer
24 views

Spectral norm of a matrix obtained by setting some entries to zero

For example can we say, that if $A$ is original matrix and $A'$ obtained from $A$ by zeroing some elements then $\|A\|_2 \geq \|A'\|_2$?
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1answer
12 views

Is there relations between earth mover's distance and vector norms?

Say I have two vectors $a$ and $b$. Can I estimate $\mbox{EMD}(a,b)$ via some combination of things like $\|a-b\|_p$ and such?
1
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1answer
25 views

Norm of the functional $f(x)=\int_0^2 x(t).(t^2-1)dt$ in the space $X=L_1(0,2)$.

I am trying to calculate the norm of the functional $f(x)=\int_0^2 x(t).(t^2-1)dt$ in the space $X=L_1(0,2)$. What we have is: $||f(x)||\le ||x||_1.||t^2-1||_{\infty}$ by extended Hölder's ...
1
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2answers
62 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 ...
1
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1answer
71 views

Upper bound on the inverse of a Grammian matrix

I have been trying to find a reasonable upper bound on the following: Given $n\in N$ and the Grammian matrix $A_n$ = (($f(i)$ , $f(j)$)) , $f(\lambda) = e^{\lambda t}$ for $0\le t \le 1$ and ...
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1answer
44 views

Induced Matrix Norm

I have trouble following a proof of the induced Norm $||\cdot||_1$ The proof can be found here: ...
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0answers
39 views

Question about functions in Sobolev space.

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$. If I consider a function $g:\mathbb{R}\rightarrow\mathbb{R}$ which has the following properties: $$ |g(x)|\leq M \qquad |g(x)-g(y)|\leq ...
5
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1answer
74 views

Is there a useful relationship between pointwise and $L^2$ distance?

It would be really convenient to get a bound on the point-wise closeness of functions by knowing their $L^2$ distance. Clearly, if two functions are close in the $L^2$ sense, you cannot get a general ...
1
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1answer
112 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. ...
0
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0answers
24 views

For what range of $a\in\Bbb R$ is $f(x)=ae^{-ax}$ in the unit ball with respect to the uniform metric on $[0,1]$

For what range of $a\in\Bbb R$ is $f(x)=ae^{-ax}$ in the unit ball with respect to the uniform metric on $[0,1]$? I.e when is $\|f\|_\infty\lt 1$ So far I see that for $a\ge 0$, ...
0
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0answers
30 views

Custom Norm Function Proof $\left \| x \right \|=\left | \xi _{1} \right |+\left | \xi _{2} \right | $

For Vector Space X consisting of ordered pairs of Complex numbers, Can we define the Norm stated below from inner product, in X ? $$\left \| x \right \|=\left | \xi _{1} \right |+\left | \xi _{2} ...
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4answers
57 views

I need help with a proof showing $\|u\|^2 = \|\operatorname{proj}_v u\|^2 + \|u - \operatorname{proj}_v u\|^2 $

So, I am dealing with the 2-norm and the projection is defined as the standard orthogonal projection, so far I have $$\|u\|^2 = \|\operatorname{proj}_v u\|^2 + \|u - \operatorname{proj}_v u\|^2 ...
0
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1answer
37 views

Derivative of $f(x)=\|Ax\|_2^2$

I'm trying to find the derivative of $f(x)=\|Ax\|_2^2$ where $A$ is some matrix and $\|u\|_2$ is the euclidean norm of $u$, $\|u\|_2 = \sqrt{u_1^2+u_2^2+\cdots+u_n^2}$ I know how to do this by ...
0
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1answer
58 views

Norm of the multiplication operator

Let $f \in L^\infty[0,1].$ It is clear that the norm of the multiplication operator $M_f : g \mapsto fg$ on $L^p[0,1]$ is $\|f\|_\infty.$ What happens in the noncommutative situation? Let us ...
0
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1answer
18 views

Natural norm for the ring $\{a+b\sqrt{2}$ | $a,b \in \mathbb Z \} $?

I am working on showing that $\{a+b\sqrt{2}$ | $a,b \in \mathbb Z \} $ is an euclidean domain. There was a similar problem showing that $\{a+b\sqrt{-2}$ | $a,b \in \mathbb Z \} $ was an euclidean ...
2
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1answer
40 views

Completely multiplicative matrix norm for certain semigroups of matrices.

I am currently working on some properties of matrix products and their norms for $\mathbb{R}^{n \times n}$ matrices and i was wondering if there exists a completely multiplicative matrix norm, i.e. ...
0
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1answer
33 views

Prove of beaing Norm function

Prove that relations beneath have conditions for being a norm function A) for $C^{n}: \left \| x \right \|= \left (\sum_{j=1}^{n}\left | \xi _{j} \right |^{2} \right )^{1/2}$ B) for ...
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1answer
27 views

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 ...
0
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1answer
28 views

derivative of 2 norm wrt matrix

I have a matrix A which is of size m,n, a vector B which of size ...
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0answers
26 views

Show that a function $\phi:(V, \|\cdot\|_{C^1}) \rightarrow (W, \|\cdot\|_{\infty})$, $f \rightarrow f'$ is continuous

I have the following: Show that the function $$\phi:(V, \|\cdot\|_{C^1}) \rightarrow (W, \|\cdot\|_{\infty}),~~f \rightarrow f'$$ is continuous. With $$\|\cdot\|_\infty = \sup\{|f(x)| ~ \big| ~x\in ...
12
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1answer
329 views

A sharper bound for $\|\cos(kA)\|_{\infty}$ for symmetric stochastic matrices

Given $A \in \mathbb{R}^{n \times n}$ that is symmetric, stochastic and diagonalizable, and $k \in \mathbb{N}$, I am interested in bounding $\|\cos(kA)\|_{\infty}$ from above. $\| \|_{\infty}$ is ...
5
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1answer
62 views

Missing a necessary power in this proof - please help.

This question is somewhat related to Gradient Estimate - Question about Inequality vs. Equality sign in one part. That question was related to part (c) of a problem I am working on, and this question ...
4
votes
2answers
52 views

$\left\{x\in H: 2\leq \|x\|\leq 5\right\}$ is compact?

In a Hilbert space $H$ of dimention infinite, $A=\left\{x\in H:2\leq \|x\|\leq 5\right\}$ is compact? (totally bounded and complete) Thanks in advance.
0
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0answers
33 views

help about supremum norm proof

T:( $C'[0,1] ,||x||₁)\to (C[0,1]$,||x||∞) Tx(t)=x'(t) for any x is in C'[0,1] $||x||₁=||x||_{\infty}+||x′||$∞ how can we prove that equality? $||x||_1 = \sum_{n=1}^{\infty}|x_{n}|$ ...
0
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1answer
49 views

Hilbert Schmidt norm inequality

I was wondering if anyone knows about an inequality for the Hilbert-Schmidt (H-S) norm of the type $|Tr (Bg)|\leq Const\cdot||B||\cdot function(||g||_{2})$ for a bounded operator $B$ and a H-S ...
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1answer
50 views

Inequality of Weighted norm

I have a question about the weighted norm inequality: The weighted norm of a vector $x\in R^{M\times N}$ is defined by: $\left \| X\right \|_{w,*} = \sum_{_{i}}\left |w_{i}\sigma _{i}\left ( X ...
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1answer
48 views

Questions over a specific case of the Muntz-Szasz theorem proof

On page 157 of this site: http://arxiv.org/pdf/0710.3570.pdf the author is proving a specific case of one direction of the Muntz-Szasz theorem. I do not understand the following 3 claims: 1) For ...
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0answers
104 views

Showing $\| Mx \|^2 = x^TM^TMx$

So, I am trying to prove $$\|Mx\|^2 =x^TM^TMx,$$ however I am running into some difficulties. Here, $M \in \mathbb{R}^{m \times n}$ and $x \in \mathbb{R}^n$. I know that when you take the transpose ...
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1answer
55 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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2answers
23 views

Bounding the distance between $L_\infty$ and $L_2$ for a continuous function

Consider a set of continuous (or even differentiable) functions $f_i(x)$, all defined for $x\in [a,b]$ for $i=1\ldots,N$. Can one define a uniform constant $c$ (which may depend on $f$) such that ...
0
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0answers
58 views

limit of p norm as p goes to 0!

Suppose we have a measure $\mu$ and a space $X$ such that $\mu(X)=1$, and a function $f \in L^r$ for some $r > 0$, where $L^r$ is defined in the usual way even for numbers less than $1$. Show ...
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1answer
62 views

Are the following norms equivalent?

We have the norms $||f||_1=||f||_\infty+||f'||_\infty$ and $||f||_2=|f(a)|+||f'||_\infty$ where $f\in C^1[a,b]$. Are they equivalent and how shoud I prove/disprove this.
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1answer
45 views

Question on the norms

I got stuck with the following (simple) question since the result I got seems to be counterintuitive: I have a function defined in terms of its Chebyshev expansion, i.e. ...
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0answers
18 views

A simple inequality with two sequences.

Suppose there are two sequences $a^n\in \mathbb{R}^p,b^n\in \mathbb{R}^d$, $n=1,2,\cdots,+\infty$. $K\in \mathbb{R}^{d\times p}$ is a matrix and let $C<\|K\|$. The following inequality always ...
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2answers
33 views

Ratio of $\|\cdot\|$ and $\|\cdot\|_{\infty}$ on $\mathbb{R}^2$

I have the following question from an old examination paper in Real Analysis: On $\mathbb{R}^2$, $\|\cdot\|_{\infty}$ is defined by ...
0
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1answer
35 views

an $L^p$ implication

Let $1<p<\infty$ and $x,y\in L^p([0,1])$ such that $\|x\|_p = \|y\|_p = 1$. Then the following implication holds: $$\left\|\frac{x+y}{2}\right\|_p=1\Rightarrow x=y\tag{*}$$ This ...
0
votes
1answer
57 views

Frechet derivative of the sup norm function on $C[0,1]$

I know that the derivative in the title does not exist for any x but I do not have a clue why. Could someone explain why this derivative DNE at 0 and I can figure out why it does not exist at x? ...
2
votes
1answer
56 views

Estimate for weak $L^{1}$ norm

Let the weak $L^{1}$ norm on $f$ be defined by $\|f\|_{\mathrm{WL}^{1}} = \sup_{t > 0}t D_{f}(t)$ where $D_{f}(t) = \mu(\{x \in \mathbb{R}: |f(x)| > t\})$ and $\mu$ is the standard Lebesgue ...
0
votes
3answers
38 views

Is this some kind of triangle inquality?

I stumbled upon the following inequality: $$\Vert x+hz-(x+y)-(p-(x+y))\Vert_2 \geq \Vert p-(x+y)\Vert_2-\Vert x+hz-(x+y)\Vert_2$$ where $p,x,y,z \in \mathbb{R}^n$. My question is: Is this some kind ...
0
votes
1answer
31 views

norm on a quotient-space

Let $M:[0,\infty)\to[0,\infty)$ be continuous and convex. Further $M$ satisfies $M(t)=0\Leftrightarrow t=0$. Let $$\mathcal L_M(\mathbb R):=\left\{f:\mathbb R\to\mathbb R \mathrm{\ measurable\ ...
3
votes
1answer
28 views

Spectral Norm of $2\times 2$ symmetric matrix

Consider a $2\times 2$ symmetric matrix, in this case, is there some closed formula for its spectral norm ? By spectral norm I mean the induced 2-norm, there is a definition here. Thanks.
0
votes
1answer
60 views

lp distance, p tends to minus infinity

Minkowki distance as p tends to minus infinity equals to the smallest difference along any coordinate dimension. Is this difference a metrics? Even if the lp norm with p tending to minus infinity is ...