Tagged Questions

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

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1answer
47 views

Let $f \in L^1[0,1]$, can we approximate $f$ by a polynomial, in sup norm?

Let $f \in L^1[0,1]$, can we approximate $f$ by a polynomial, in sup norm ? I know that the algebra of polynomials is dense in algebra of continuous functions, wrt to sup norm, And I know that if $f ...
2
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2answers
80 views

Comparable norms on the space of polynomials?

Are the norms: $$\|P\|_1=\int^1_0\|P(t) \| dt\mbox{ and }\|P\|_2=\sup_{0\le t\le1} |P(t)|.t$$ comparable on the vector space $X=\mathcal{P}[0,1]$, i.e. all polynomials on $[0,1]$? Here, i try to ...
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1answer
84 views

Prove that the operator norm is a norm

Exercise: Prove that the operator norm of the set $S$ of all linear operators $L:R^n\to R^m$ defines a norm on $S$ Definition of norm: A positive function $\| .\|$ on a real vector space $V$ is a ...
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0answers
25 views

The norm of a vector with gaussian noise

Say I have a vector of length n, $v \in R^n$ where $0<=v(i)<=1$ for each i, Now, I add noise: let $n$~$Normal(0,\sigma)$ And to each i I add noise v(i)+n , such that the noises are independent ...
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0answers
26 views

Why this relation in Hilbert space (with inner product $< >$) holds?

$c_k$, $f_k$ are sequences in Hilbert space, $g$ is a function. Why this relation below holds? How you derive it? $\sum_k|c_k\left \langle f_k,g \right \rangle|\leq(\sum_k|c_k|^2)^{1/2}(\sum_k|\left ...
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2answers
37 views

Why $||f||=\mathrm{sup}_{||g||=1}|\langle{f,g}\rangle|$

My question is why $||f||=\mathrm{sup}_{||g||=1}|\langle{f,g}\rangle|$? for any two functions $f$ and $g$ with $||g||=1$, and $||\;||$ denotes the 2-norm. I have tried to use the triangle inequality ...
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0answers
35 views

Conditions on $\alpha_n$ for $n_\alpha(x)=\sum^\infty_{n=1} \alpha_n\vert x_n\vert $ to be a norm on $l_p$

When $n_\alpha(x)=\sum^\infty_{n=1} \alpha_n\vert x_n\vert $ is a norm in $\mathcal{l}_p=\lbrace (x_k)^\infty_1 : \sum\vert x_k\vert ^p \lt\infty\rbrace $ and $\alpha\in\omega$. and $\omega$:space of ...
0
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1answer
57 views

If A is positive-definite what can we say about $x^t A y$ or $y^t A x$?

If $A$ is an $n \times n$ positive definite matrix in $\displaystyle f(x) = \frac{\sqrt{x^t A x}}{2}$, can I claim that $f(x+y) \leq f(x) + f(y)$?
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2answers
33 views

Relation between the weighted matrix norm and the weights

For a nonsingular matrix $W \in \mathbb{C}^{m\times{}m}$, the weighted vector norm is defined as $||\overrightarrow{x}||_W = ||W\overrightarrow{x}||$. Let $||A||$ denote the induced matrix norm by the ...
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2answers
80 views

Can anyone explain this isometry to me? $T: (\mathbb R^n,\|\cdot\|_1)^*\to (\mathbb R^n,\|\cdot\|_{\infty})$, $ T(x)(y) = \sum_{i=1}^n x_i y_i$

Can anyone explain this isometry to me? $$T: (\mathbb R^n,\|\cdot\|_1)^*\to (\mathbb R^n,\|\cdot\|_{\infty}),\qquad T(x)(y) = \sum_{i=1}^n x_i y_i$$ I don't get what the domain and image of $T$ are. ...
0
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1answer
64 views

proofread $\left \| f -g\right \|_{p}^{p}=p(p-1)\int_{0}^{\infty}\int_{0}^{t}[m(f>t/g>s)+ m(g>t/f>s)]|t-s|^{p-2}dsdt$

Is this argument correct? $\left \| f -g\right \|_{p}^{p}=$ $\int_{0}^{\infty}m(|f-g|>t^{\frac{1}{p}}) dt\stackrel{c.o.v.}{=}$ $p\int_{0}^{\infty}m(|f-g|>t) t^{p-1}dt\stackrel{t=|u-s|}{=}$ ...
1
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1answer
26 views

Can we find equivalent norms for ordinary numbers $m$ and $M$ s.t. $0<m<M$?

Let $\|\cdot\|$ be a norm and $0<m<M$. Can we say that we can find another norm $\|\cdot\|_1$ s.t. $\|\cdot\|$ and $\|\cdot\|_1$ are equivalent with respect to numbers $m$ and $M$?
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1answer
38 views

Is the norm $\|A\|_\mathrm{op}$ for a matrix $A$ a Euclidean norm?

$\displaystyle \|A\|_{\operatorname{op}} = \sup \frac{\|Av\|}{\|v\|}$ for $v$ in $V$, where $\|v\|$ is a Euclidean norm of a vector $v$ in a vector space $V$. I saw a counter-example that was based ...
2
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1answer
66 views

For a matrix $A$, is $\|A\| \leq {\lambda}^{1/2}$ true?

In class I saw a proof that went something along these lines: Define $\|A\| = \sup \dfrac{\|Av\|}{\|v\|}$ for v in V, where the norm used is the standard (Does this even exist?) Euclidean norm in V. ...
1
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1answer
60 views

Computing H1 norm numerically

I'm solving a PDE numerically using FDM and Spectral Methods. I understand how to compute the $L_{2}$, but I dont understand how to compute the $H_{1}$ norm. What does the $u'$ mean in the below ...
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2answers
86 views

let $A$ be an n by n matrix, show that $||A||_{OP} \leq ||A||_{HS} \leq \sqrt{n} ||A||_{OP}$

We are given $A \in M_{n}(\mathbb R)$ and the following norms: $||.||_{e}$ is the standard euclidean norm of $\mathbb R^n$. $||A||_{OP}$ is the operator norm of $A$, meaning $||A||_{OP} = ...
1
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1answer
42 views

Proof check for completeness

I'd like to see if the proof I have is adequate. Statement. Let $X$ and $Y$ be Banach space, the product $X\times Y$ is a vector space under coordinate operations with norm $$ \|(x,y)\| = \|x\|_X ...
1
vote
1answer
37 views

An equation related to covariance matrix, square root of the matrix, and Euclidean norm.

How can I prove this equation: $${ ({ x }^{ T }\Sigma x) }^{ 1/2 }={ \left\| { \Sigma }^{ 1/2 }x \right\| }_{ 2 }$$ In which $\Sigma $ is a covariance matrix. I tried some numerical examples in ...
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1answer
54 views

When sup norm,i.e. $\| x|| = \sup|x(t)|$ for $\forall t\in T $in C[0,1] for $T \subsetneq [0,1]$

When sup norm, i.e. $\| x|| =\sup|x(t)|$ for $\forall t\in T $in C[0,1] $T$ is such that $T \subsetneq [0,1]$. What condition should be applied to $T$ to make $\| x||$ a norm. I cannot show for ...
2
votes
1answer
45 views

Continuous functions. Second norm

Let $f:[a,b]\rightarrow \mathbb{R}^{d}$ be continuous. I need to prove that $\left \| \int_{a}^{b}f(x)dx \right \|_{2}\leq \int_{a}^{b}\left \| f(x)) \right \|_{2}dx$.
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1answer
26 views

Analytical solution to fitting two functions

I have two oscillatory functions $f(x)$ and $(k x)^2 g(x)$ where $f$ and $g$ are known and it is also known that the two functions are approximately similar. How can I analytically find the best ...
0
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1answer
30 views

Example for a two norms and vector space which are equivalent on this Vectorspace

I need an example where the two norms and the vector space are equivalent on the vector space.
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0answers
38 views

Pareto distribution and matrix

I am wondering if there are any bounds are known on the eigenvalues of random matrix whose entries are with Pareto distribution? Thank you.
2
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1answer
210 views

Converting sum of infinity norm and L1 norm to linear programming

So I'm trying to convert this minimization problem, min $\parallel Ax-y \parallel_{\infty}$ + $\parallel x \parallel_{1}$ where $A$ is $m$ by $n$, $y$ is $m$ by $1$ and $x$ is $n$ by $1$. into a ...
1
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1answer
62 views

question about norms and convex set

Suppose $\overline{B}(0;1) = \{ x \in X : ||x|| \leq 1 \}$ is the closed unit ball on a vector space $X$. MY question is: is the following true? If $\overline{B}(0,1) $ is not convex, then $|| \cdot ...
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3answers
61 views

Norm of vectors inequality

I tried proving this with triangular inequality but i was not right can any one help me with this
0
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1answer
51 views

$l^p$ is super subset of $ l^q$ if and only if?

$$ l^p \subseteq l^q. $$ if and only if ? $l^p$ is a subset of $l^q$ when is it possible ? !
5
votes
2answers
61 views

Given a matrix $A$ such that $||A||<1$, prove that $I-A$ is invertible

Note: This is a question seen in class while discussing metric spaces and norms, so my recollection might not be 100% accurate. I saw a proof in class, but I wanted to know if there was a different ...
0
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1answer
55 views

Inequality matrix norm

Let $A$ be an $n\times n$ random matrix $A=rand(n,n)$. Let $\alpha=max_{i,j}|a_{ij}|$ (i.e, $\alpha$ is the largest entry in $A$ in absolute value).I need to show that $\ \alpha < \| A \|_{2}$. ...
2
votes
1answer
35 views

Norm of vector with respect to operator

Define $L$ is a linear operator maps from $E^n$ to $\mathbb{R}$, its norm is defined as $||L||_{op}=\sup\limits_{||x||=1}L(x)$, where $||\cdot||$ is any norm on $E^n$. How to show that ...
5
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0answers
118 views

renorm a Banach space to make an operator have spectral radius equal to norm

Let $X$ be an infinite-dimensional complex Banach space equipped with the norm $\lVert\cdot\rVert$, and let $T\in\mathcal{L}(X)$ a bounded linear operator on $X$. Let $r(T)$ denote the spectral ...
0
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1answer
35 views

Matrix norm of product equal implies equality in norms of factors

Given a matrix $A$, if $$\|Av\|_1=\|Aw\|_1$$ for given vectors $v$ and $w$, then does $\|v\|_1=\|w\|_1$? Here $\|\,\cdot\,\|_1$ denotes the $L^1$ norm.
3
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1answer
62 views

Hilbert-Schmidt norm/smooth manifolds

Given two riemannian manifolds $M$ and $N$ and a smooth map $f$ : $M$ $\rightarrow$ $N$, we define the energy density of $f$ as the smooth function $e(f)$ : $M$ $\rightarrow$ $\mathbb{R}$ given by ...
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0answers
37 views

How to prove that $\ell_p$ norm is smaller than $\ell_q$ norm if $p > q$?

Let $x \in \mathbb{R}^n$ be an $n$-dimensional vector. It is apparently well-known that, \begin{equation*} \Vert x \Vert_p \le \Vert x \Vert_q \end{equation*} for any $p > q \ge 0$. (cf. ...
2
votes
1answer
85 views

How can we replace $\sup$ with $\max$ in definition of subordinate norm for finite-dimensional vector space?

I have been given the definition of a subordinate (operator or matrix) norm: $$\lvert\lvert A \rvert\rvert=\sup_{x\in V}\frac{\lvert\lvert Ax \rvert\rvert}{\lvert\lvert x \rvert\rvert},$$ where $V$ is ...
4
votes
3answers
68 views

Why does $\|\sum_{i=1}^{N}\langle f,\phi_i\rangle\phi_i\|^2 = \sum_{i=1}^{N}|\langle f,\phi_i\rangle|^2 $?

I'm reading about Fourier analysis and there is one equality, which I don't understand. Why does: $$\left\|\sum_{i=1}^{N}\langle f,\phi_i\rangle\phi_i\right\|^2 = \sum_{i=1}^{N}|\langle ...
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1answer
40 views

In a normed space, is it always true that $\|a_1e_1+\dots+a_ne_n\|\geq |a_i|\|e_i\|$?

In a normed space, is it true in general that $\|a_1e_1+\dots+a_ne_n\|\geq |a_i|\|e_i\|$ for all $1\leq i\leq n$? $e_i$ are basis elements of the vector. This is definitely true for the Euclidean ...
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0answers
19 views

Estimation of $\alpha_{0}$ using norm of $y_{0}$ and $y_{1}$

If $$y_{0}=\frac{1}{5}x^{5}+\frac{1}{2}x^{4}+\frac{2}{3}x^{3}+x^{2}+2x+1$$ and ...
1
vote
1answer
22 views

Verifying whether a given function can be a norm.

I was asked to prove that given the vector space $\Bbb{R}\times\Bbb{R}$, the function $f(p)=(\sqrt{a}+\sqrt{b})^2$, where $p=(a,b)$, does not define a norm (on $\Bbb{R}\times\Bbb{R}$). Is the ...
3
votes
1answer
145 views

Norm- Orthogonal Projection $A^{t}A=I_{n}$

Let $A_{m\times n}$ be a matrix such that $m\geq n$ and $A^{t}A=I_{n}$. It is given that the columns of $A$ are orthonormal. I need to show that the 2-norm of each row of $A$ is $\leq 1$. I have no ...
2
votes
1answer
46 views

Weak Lower Semicontinuity Generalized to any $L_{p}$ space

I am having difficulty with the following proof: Generalize the weak lower semi-continuity of$L^{p}$ norms to all $1\leq p < \infty$; i.e., show that if $u_{n}\to u$ weakly in $L^{p}$, then ...
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1answer
73 views

Linear algebra - question about vector norm and eigenvalues

Maybe a basic question, but I'd like to know the reasoning behind it if its true. suppose I have a matrix $A \in \mathrm{Mat}_n(\mathbb R)$ with the eigenvalues $\lambda_1 ,\lambda_2 ,..., ...
0
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1answer
39 views

Integral of a function is not affected by altering the function values at zero-measure set

I'm studying about Fourier analysis from a book Fourier analysis and its applications, Folland 1992 and I have one point in the source I need clarification about: On page 69 it is stated that: "The ...
1
vote
0answers
114 views

About $f(s)=\sum_{a^2+b^2>0} \frac{(-1)^{a^2+b^2}}{(a^2+b^2)^s}=0$ and the Extended Riemann Hypothesis.

Let $s$ be a complex number with a strictly positive real part ($Re(s)>0$). Let $f(s)=\sum_{a^2+b^2>0} \dfrac{(-1)^{a^2+b^2}}{(a^2+b^2)^s}$ where the sum runs over all positive integers $a,b$ ...
0
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0answers
18 views

The spectral norm of $ \begin{bmatrix} F_{k+1} x & F_{k}\\F_{k} x^2 & F_{k-1} x\end{bmatrix} $

In mathematics, the matrix induced norm is a natural extension of the notion of a vector norm to matrices. In the special case of $ p = 2$ (the Euclidean norm) and $m = n$ (square matrices), the ...
0
votes
1answer
22 views

What happens to $l_1$ if i change coordinate system.

Let $x =(x_1,\ldots,x_n) \in \mathbb{R}^n$ and also $x=\sum_{i=1}^m t_i u_i,$ where $t_i \in \mathbb{R}$ and $u_i \in \mathbb{R}^n.$ Is it true that $||x||_1 \geq \sum_{i=1}^m |t_i|$ ?
0
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0answers
47 views

is there any infinity norm bound to simplify this

I have a problem of the form $$\sup_{x\in\Bbb{C}^n}\left\{\frac{\|Ax\|_\infty}{\|Bx\|_\infty}\right\}$$ where $A$, $B$ are matrices with different number of rows and $x$ is an $n$ dimensional vector. ...
2
votes
1answer
69 views

bound on trace-norm of product of matrices

Is it true that $$ \|ABA^\dagger\|_1\leq \|A\|^2\|B\|_1, $$ where $\|A\|_1$ is the trace norm, $\|A\|$ is the spectral norm, and $A$ and $B$ are square matrices?
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0answers
40 views

Solving a linear matrix equation with respect to the maximum of the euclidian distances between rows.

With $n>m$, real number matrices $A$, $B$, $C$ are shaped like: $$A=\left( \begin{array}{ccc} a_{1,1} & \cdots & a_{1,m} \\ \vdots & \ddots & \vdots \\ a_{n,m} & \cdots ...
0
votes
1answer
12 views

when linear mapping keeps monotonicity of $L_2$ norm

Consider an arbitrary vector $\alpha$ from vector space $R^p$, a linear mapping $A: \alpha\rightarrow A\alpha$ transforms $\alpha$ to $A\alpha$ in space $R^q$. What condition should $A$ satisfy so ...