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

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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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23 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 ...
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1answer
32 views

bounded subset of normed space

Suppose $X$ is a normed linear space and $S\subset X$. Show that if $$\sup_{x\in S}\{\mid f(x)\mid\}<\infty$$ for every $f\in X^{\ast}$, then $S$ is a bounded in norm.
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23 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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30 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.
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57 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 ...
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1answer
41 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
53 views

Norm of vectors inequality

I tried proving this with triangular inequality but i was not right can any one help me with this
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1answer
42 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 ? !
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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
27 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
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1answer
28 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 ...
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83 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 ...
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1answer
24 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.
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45 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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25 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
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1answer
40 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 ...
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3answers
64 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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2answers
35 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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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 ...
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1answer
18 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 ...
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1answer
38 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 ...
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1answer
39 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
59 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 ,..., ...
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1answer
33 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 ...
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101 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 $a,b$ ...
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14 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 ...
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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 $||x||_1 \geq \sum_{i=1}^m |t_i|$ ?
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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
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1answer
37 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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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
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1answer
11 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 ...
3
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3answers
60 views

whats the difference between $|v|$ and $||v||$?

$v$ being a vector. I never understood what they mean and haven't found online resources. Just a quick question. Thought it was absolute and magnitude respectively when regarding vectors. need ...
3
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2answers
96 views

Question about sup norm

Let $x \in \mathbb{R}^n$. Define $|x| = \max\{ |x_1|,...,|x_n|\} $. I want to show that this is a norm on $R^n$. This is my reasoning. First, notice $$ |x| = \max\{ |x_i| \} \geq |x_i| \; \forall i ...
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1answer
23 views

Norms in $\mathbb R^2$ - Strategy to prove that a norm is a norm on a set.

What points should I prove when I am asked to prove that a particular norm, say $||(x,y)||=(|x|^{1/2}+|y|^{1/2})^2$, is a norm in $\mathbb R^2$? P.S I have read about the difference between a metric ...
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2answers
55 views

How to calculate the norm?

We have the vector : $$ w=(1,3,5,1,3,5,\ldots,1,3) \in \mathbb{R}^{3k-1}, $$ and we want to calculate its norm $\|w\|$. Now I would like to know how the norm $\|w\|$ can be calculated.
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1answer
35 views

An inequality on matrix norm

Does inequality $\|A\|_2\leq \| |A|_m \|_2 $ hold for all square matrices $A$ ? Where $|A|_m$ is also a square matrix, defined as $|A|_m := [|a_i,j|]$. Two examples are provided for the case that ...
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1answer
29 views

For general non-symmetric square matrices is there a matrix norm that is invariant under similarity transformations?

I think that there is no similarity-invariant matrix norm for general matrices. But are there similarity invariant norms for special types of matrices (e.g. for matrices whose eigevalues are different ...
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51 views

norm of differential operator on $P^n[0,1]$

Consider the space $P^n[0,1]$ of polynomials of degree $\leq n$ on $[0,1]$, equipped with the sup norm. Now, this is a finite dimensional space, so all linear operators have to be continuous, hence ...
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What is the correct notation for defining norms in measure spaces?

For a function $f \in L_2(R)$, we can define its norm as $$ \|f\|_2^2 = \int f^2(x) dx $$ If I use a different measure $\mu$, I can in turn define the norm as $$ \|f\|_2^2 = \int f^2(x) \mu (x) dx ...
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1answer
60 views

max induced norm of matrix

I have to prove that matrix norm $||A||_\infty$ induced by vector norm $||x||_\infty = \smash{\displaystyle\max_{1 \leq k \leq n}} |x_k|$ where $x_k$ is k-th element of vector can be disribed by ...
2
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1answer
58 views

Any positive linear functional $\phi$ on $\ell^\infty$ is a bounded linear operator and has $\|\phi \| = \phi((1,1,…)) $

This is a small exercise that I just can't seem to figure out. When I see it I'll probably go 'ahhh!', but so far I haven't made any progress. I'd like to prove that any linear functional $\phi$ on ...
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1answer
66 views

comparison of 3 topologies on C[0,1]

I have a ring of continuous functions from $[0,1]$ to $\Bbb R$. And two norms $C[0,1]\to\Bbb R$. One is supremum of $|f(x)|,$ the other the value of $\int_0^1|f(x)|$. Then I get a Cartesian product of ...
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1answer
33 views

Showing that a map defined using the dual is a bounded linear operator from X' into X'

I have trouble answering the second part of the following exercise. Any help would be appreciated! Let $(X, \| \cdot \|)$ be a reflexive Banach space. Let $\{ T_n \}_{n = 1}^\infty$ be a sequence of ...
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how to prove that the limit of this sequence of functions is continuous?

I have a norm that works in function space and that is $‖∙‖_{sup}:C([0,1])→R$, $‖∙‖_{sup}:=sup${$|f(t)|$}. I need to show that the metric is complete. So I need to show that every Cauchy sequence of ...
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2answers
42 views

Is convergence in the norm equivalent to convergence of norms?

If $\| \cdot \|$ is a norm on some space. Does the equivalence go both ways? $$\| f_n-f \| \to 0 \Longleftrightarrow \| f_n\| \to \| f\| $$ The $\implies$ direction is obvious since $\| f_n-f \| ...
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1answer
51 views

Integral Operator Theory on $L^2[0,1]$

Let K be the integral operator on l^2[0,1] defined by itex(t) = \int_0^t (t-s)f(s)\,ds[/itex] where 0\leq t\leq 1 Show that ||K|| <1 and that tex(t)= \int_0^t ...
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52 views

about vectors norm

in the following article http://blanche.polytechnique.fr/~mallat/papiers/MallatPursuit93.pdf page 3 he say: $$y= \langle y , a_{k_0} \rangle a_{k_0} + R $$ with $a_{k_0}\in D$ with $\forall ...
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1answer
69 views

addition and multiplication of functions in function space, continuous?

I have a norm that works in function space of C[0,1]. How do I show that addition and multiplication of functions (C[0,1]xC[0,1]->C[0,1]) are continuous functions?
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26 views

Injective norm on tensor algebra of a finite-dimensional Banach space

Suppose that $V$ is a finite n-dimensional Banach space, and suppose that $T(V)$ is the tensor algebra on it. Furthermore, suppose that $T^{(n)}(V)$ is the "abridged" tensor algebra obtained by taking ...