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

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63 views

Under what condition does $\int_a^b|f(x)|=0$ imply $f=0$? ( at least a.e.).

I'm working through Real Analysis by Royden & Fitzpatrick, and on the first section on $L^p$ spaces they always skip the last property of norms without much commentary. Namely, that $||f||=0 \iff ...
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1answer
43 views

conjugate transpose of contraction

is it true, for any matrix $T \in \mathbb{C}^{n \times n}$ (with scalar product $\langle .,.\rangle$ and the norm $\left\Vert v\right\Vert = \langle v,v\rangle^{\frac{1}{2}}$), that, if $T$ is a ...
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2answers
21 views

$\not\exists$ number $c\geq 0$ for which $||f||_{max}\leq c||f||_1$. But $\exists$ $c\geq 0 $ for which $||f||_1\leq c||f||_{max}$

I'm reading through Real Analysis by Royden & Fitzpatrick and I always try to deal with the first few problems of each section. For the first section on $L^p$ spaces there were mostly routine ...
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4answers
45 views

Derivative of matrices product

Find the derivative of the following matrix $ f(X) = a^TXb, $ where $ a,b ∈ R^n $ and X is an n×n matrix. Please give me some serious hint!
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1answer
37 views

If $f_n\to f$ in the $L^1$ norm, show that there is a subsequence $f_{n_k}$ which converges a.e. to $f$.

This question is from a problem set on $L^p$ spaces in my undergraduate-level real analysis course. I said that $f_n$ converges if and only if it is Cauchy. Therefore, $\exists \, N\in\mathbb{N} \; \...
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2answers
83 views

Does $\{\sin (nx)\}_1^\infty$ converge in the $L^1$ norm on $[0,2\pi]$?

This is a homework question from a problem set in an undergraduate-level real analysis course (coming from merely an intro to analysis course) about $L^p$ spaces. Show that $\{\sin (nx)\}_1^\...
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1answer
22 views

Derivates of a vector in respect to the elements

Find the derivative of $(a) f(x)= \frac{1}{x_3} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ $(b) f(x)=trace(xx^T)$ where $ x_1 , x_2, x_3 $ are the first three elements of x.
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1answer
63 views

Jacobian of second norm

Find the Jacobian of the following function: (a) $f(x)= \|x -x_0 \|_2$ (b) $f(x)= \log(\|x \|_2)$ Please give me some serious hint!!
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votes
1answer
27 views

Derivative of a nonsingular matrix

Show that : $$\frac{d}{dt} A^{-1}(t) = -A^{-1}(t) (\frac{d}{dt} A(t) ) A^{-1}(t) $$ A(t) is a matrix.
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2answers
68 views

Cauchy Sequence in Normed Space

Let $(E, ||\cdot ||)$ be a normed space and let $(x_n)$ be a sequence in $E$. Show that the following conditions are equivalent: (a) $(x_n)$ is a Cauchy sequence. (b) For every increasing function $...
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1answer
20 views

Proof for Semidefinite symmetric matrix product

If A is symmetric positive semidefinite, show that: (a) For any matrix B,$ BAB^T $is also positive semidefinite. (b) All the diagonal elements of A are nonnegative.
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24 views

Minimization using Singular value

Let $A$ be a $p\times q$ matrix, with rank $q$. Show that the vector $x$ that minimizes $\|Ax\|_2$ under the constraint $\|x\|_2 = 1$ is the right singular vector of $A$ corresponding to the smallest ...
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2answers
29 views

How to maximize matrix products

Find a unit vector v1 and a unit vector v2, such that the term: $$v^T \begin{bmatrix} 6 & -2 \\ -2 & 6 \end{bmatrix}v$$ is minimized and maximized, respectively. What are the minimum and ...
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2answers
27 views

Multiplication of an orthogonal matrix by its first column

I am given a real orthogonal matrix Q (nxn), where the first column of Q is the vector x (nx1) where the 2-norm of x equals 1. I am asked to prove that QTx has first entry 1 and all the others zero: ...
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3answers
394 views

Norms and equivalence classes question

Let $f\in C[0,1]$. Recall two of the norms we considered in class: $$\|f\|_\infty = \sup_{t\in[0,1]}|f(t)|, \quad \|f\|_1 = \int_0^1|f(t)|\ \mathsf dt. $$ Consider the space $C^1([0,1])$ of ...
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1answer
33 views

lower bound on the minimum singular value of $\underline{\sigma} (A+B)$

what can we say about the lower bound on $\underline{\sigma}(A+B)$? Can we say the following? $\underline{\sigma}(A+B)>\underline{\sigma}(A)+\bar{\sigma}(B)$, where $\underline{\sigma}$ denotes ...
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0answers
34 views

Can every polynomial be represented as a field norm?

Consider $f\in\mathbb Q[x_1,x_2,...,x_n],$ such that its degree is n and is irreducible.Can we find a normal extension $K/Q$ of dimension $n$ such that $$N_{K/Q}(\alpha)=f?$$
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1answer
100 views

Relation between infinity norm and LU factorization

Let $A$ be a non-singular $n \times n$ matrix and suppose that Gaussian elimination with partial pivoting has been applied to produce $PA = LU$, where $P$ is a permutation, $L$ is a unit lower ...
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0answers
39 views

Find the distance between two convex sets

Let's say that ||.|| is an Euclidian norm in $R^3$ and we have two sets in $R^3$ defined by inequalities: $Y = \{y| f(y)<a\}, Z = \{ z| g(z)<b \} $ Let's say that $f$ and $g$ are convex and ...
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0answers
93 views

Variational characterization of nuclear norm

The nuclear norm $||\cdot||_{*}$ of a matrix is defined as the sum of its singular values. Working from the result at the bottom of this blog post, we have, for a matrix $\mathbf{X}$ and its ...
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0answers
20 views

Do $\ell_{\infty}$ and $\ell_{2,1}$-norms give similar results? and why?

I am aware that $ \ell_{2,1} $-norm is being used for inducing a structure in the sparse matrix. I was told by someone that $ \ell_{\infty} $ and $ \ell_{2,1} $-norms give similar results. But how and ...
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0answers
19 views

What's a name of this quantity: $\max_i \big\lvert {\| x_i \|_2}^2 - 1 \big\rvert$?

What's a name of this quantity: $\max_i \big\lvert {\| x_i \|_2}^2 - 1 \big\rvert$? I defined this quantity to measure how the given set of real vectors is far from a set of normalized ones. Perhaps ...
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0answers
18 views

Is this map monotonic? (PDEs and functional analysis)

Let $\Omega$ be a bounded domain and let $g \colon \mathbb{R} \to \mathbb{R}$ be a monotonic increasing and continuous function. Given any $u, v \in L^2(0,T;L^2)$, does it follow that $$\int_0^T \...
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1answer
42 views

How to prove triangle inequality for $\|u\| = \langle{u,u}\rangle ^{0.5}$

I want to prove that $\|u\| = \langle{u,u}\rangle ^{0.5}$ satisfies the 4 conditions for being a norm. I've already proved the first 3 conditions but I'm stuck on the triangle inequality i.e. $$\|x + ...
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2answers
38 views

A bounded operator satisfies $\|Tx\|\leq\|T\| \|x\|$

I want to prove that if $T$ is a bounded linear operator, from $X$ to $Y$, then for each $x\in X $ we must have $\|Tx\|\leq\|T\| \|x\|$. Let's take some nonzero $x\in X$ . Then $\frac{\|Tx\|}{\|x\|} ...
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1answer
31 views

A lower bound on the form of the resolvent operator

Let $A\in\mathbb{C}^{n,n}$ and $x\in\mathbb{C}^{n}$, $\|x\|=1$. Is there any $c(z)>0$ such that $$|\langle x, (A-z)^{-1} x \rangle|\geq c(z), \quad \text{ for } |z|>\|A\|\,?$$ Recall that it is ...
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1answer
19 views

mass concentration inequality for polynomials

I am trying to prove the following: Let $p$ be a polynomial of degree n and let $I=[0,1]$ and $E\subset I$ a measurable set of non-zero measure, i.e., $\mu(E)\neq 0$. Then, $$\sup_{x\in I}|p(x)|\leq \...
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4answers
89 views

What is the meaning of $||x||=\sqrt{\langle x,x\rangle}$

I understand that a norm assigns a length to each vector in a vector space. I have been told that $$||x||=\sqrt{\langle x,x\rangle}$$ is a norm. So does this equation find the length of vector $x$, ...
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1answer
25 views

Real analysis about open sets in norm metric space

Hi guys I'm having trouble starting this problem.. Consider the metric space $\mathbb R^2$ with the usual, $∥ \cdot ∥$-based metric. Show that, for any real number $r$, the set $\{(x_1, x_2) \in \...
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0answers
52 views

The dual norm of a operator matrix norm

Let as look at matrices $B$ in $\mathbb{R}^{p\times q}$ together with the following operator norm: $$||B||_{op}:=\max_{\beta}\frac{|B\cdot \beta|_{p}}{|\beta|_{q}}.$$ Here $|\cdot|_{p}$ is any norm on ...
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1answer
32 views

Hahn-Banach separation theorem with a countable subset of functionals

For a separable Banach space $X$, the unit sphere of $X^*$ always contains a countable set $D$ such that $$ \left\Vert x \right\Vert = \sup_{f \in D} \left\vert f(x) \right\vert \qquad \mbox{ for ...
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3answers
46 views

Is there a solution to $Tx=y$ such that $\|x\|$ is minimal?

I recently used the method of least norms to solve an underdetermined system of linear equations for a problem at work. This got me thinking, if I were to think about this more generally, does such a ...
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0answers
26 views

The function is a norm on $X$

Show that if $(X, \langle, \rangle)$ is a space with inner product then the function $x \mapsto ||x||:= \langle x, x \rangle^{\frac{1}{2}}$ is a norm on $X$. I have tried the following: From the ...
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1answer
17 views

Function norm proof using two-norm

I am trying to prove that $||f + g|| ≤ || f || + || g ||$ holds for the two-norm function norm. Two-norm: $||f||_2 = (\int_a^b f(x)^2 dx)^{1/2}$ $||f||_2 = (\int_a^b (f(x) + g(x))^2 dx)^{1/2}$ $||f|...
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0answers
20 views

Norm squared imaginary number

I have a function $f_0 = \sum_i b_i c_i$ where $b_i$ are orthonormal functions and $c_i$ are constants and we calculated the following thing: ${||f-f_0||}^2 = {||f||}^2 + \sum_j {|c_j|}^2 - \sum_j ...
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1answer
37 views

How to solve $\max|x_1|\Sigma_{i=1}^n|x_i|$, with constraint $\Sigma_{i=1}^n|x_i|^2=1,n>2$

Alternatively, that making the upper bound for $|x_1|\Sigma_{i=1}^n|x_i|$ as tight as possible is also welcome. For me, if $\Sigma_{i=1}^n|x_i|=\sqrt{n}$, then $|x_1|=\sqrt{1/n}$. If $|x_1|=1$,then $\...
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0answers
29 views

Why is the *least-norm* solution useful or desirable?

For a problem at work, I had to write a program that would solve a system of linear equations. Often times the problem would lead to an underdetermined system (where there are less equations than ...
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2answers
116 views

About Cauchy–Schwarz inequality

For the vectors $x$ and $y$, the Cauchy–Schwarz inequality reads $$ |x\cdot y|\leq||x||\cdot||y|| $$ Does this inequality only hold for 2-norm? Or for any norms? Thanks in advance.
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Proving that this is a particular norm

Consider the function $||\cdot||_\infty:l^2\to[0,\infty)$ given by $||(a_n)||_\infty = \sup\{|a_n|:n\ge 1\}.$ So I want to prove that this is indeed a norm. I did the following, we note that if $(...
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28 views

Definition of the norm of a function in $L^2(\mathbb{R^n})$?

Have I got the following correct - Let $f \in L^2(\mathbb{R^n})$, with $n = 3$ for the sake of example. Is the the norm of $f$ in this space as follows? $$||f||_{L^2(\mathbb{R^3})} = (\int_{\mathbb{R^...
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1answer
146 views

Is $\sup_{\| f \| \leq 1}{\left| \int f d\mu \right|} = \sup_{\| f \| \leq 1}{\{ |\mu(f)| \}}?$

The answer given by t.b. mentioned the following One of the most convenient way of writing a total variation norm is $$\| \mu \| = \sup_{\| f \| \leq 1}{\left| \int f d\mu \right|}$$ In ...
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2answers
31 views

Orthogonality of limits in arbitrary inner product space

This is a question I'm having quite a bit of trouble with. Let $(X,\langle\dot{},\dot{}\rangle)$ be an inner product space and $\{{x_n}\}$ be a sequence in $X$ that converges to some $x\in X$ in the ...
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1answer
55 views

$L^p$ and $L^1$ norms

As $L^p$ norm of $f:\mathbb R\to \mathbb R^{n}$ is defined as $$ \|f(t)\|_p:=\left(\int_S\|f(t)\|^pdt\right)^{1/p} $$ I have two questions: What kind of norms for the function $f(t)$ in the ...
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17 views

Is this a property of Normed Vector Spaces?

Let $X$ a normed vector space on $(\mathbb{K}, + , . )$. Is the following assertion true? Any $x$ of $X$ can be written as $x = \alpha a$ , $\alpha \in \mathbb{K}$ , $a \in X$ with $||a||_X=1$ ...
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1answer
49 views

Are the infinity norm and the the induced norm equivalent in $l^2$?

Consider the function $||\cdot||_\infty:l^2\to[0,\infty)$ given by $$||(a_n)||_\infty = \sup\{|a_n|:n\ge 1\}.$$ So I want to know if this norm is equivalent to: $$||(a_n)||=\sqrt{\sum |a_n|^2}$$ ...
0
votes
1answer
59 views

Open Ball and Lipschitz Equivalence equivalence

I am trying to show that two norms $\|\cdot\|$ and $\|\cdot\|^\prime$ are Lipschitz equivalent if and only if there exist numbers $r,R >0$ such that $B_r \subseteq B_1^\prime \subseteq B_R$ where ...
2
votes
0answers
23 views

Lower bound on multiplicative norm of Banach algebra.

Let $A$ be a Banach algebra over $\mathbb{C}$ and $N: A \to \mathbb{R}_{\ge 0}$ the corresponding multiplicative norm. For $a \in A$, do we have$$\limsup_{n \to \infty} (N(a^n))^{1\over{n}} \le N(a)?$$...
0
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1answer
31 views

Geometry of a Cauchy sequence in a normed space

A sequence in a normed space $X$ is called a Cauchy sequence if and only if for every $\epsilon > 0$ there exists an integer $N\in \Bbb N$, such that $\|x_n-x_m\|\lt \epsilon$ for all $n>m>N$....
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1answer
14 views

Deducing equivalence between norms from simple condition

Let $||\cdot||,\;||\cdot||'$ be norms on $V$. Suppose for some $a,b>0$ we have: $$||x||<a\Rightarrow ||x||'<1\Rightarrow ||x||<b$$ Show that $||\cdot||,\;||\cdot||'$ are Lipschitz ...
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0answers
17 views

Understanding the Quotient norm, constructed example

I just read the definition of the quotient norm, but it had no examples, so I constructed my own. Is this correct? Let $(X=\Bbb R^3,\|\cdot\|)$ with Euclidean norm and $Y\subset X$, $\quad Y=\{(a,0,0)...