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
42 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
votes
2answers
38 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
votes
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
281 views

How to prove that the spectral radius of a linear operator is the infimum over all subordinate norms of the corresponding norm of the operator.

I am trying to understand a proof I have seen of the following theorem: $$\rho(A)=\inf_{\|\cdot\|}\|A\|.$$ I understand that to do this, the idea is to show that 1) $\rho(A)\leq\|A\|$ for any norm, ...
0
votes
2answers
157 views

Matrix norm proof

Given is $\left | \left | A \right | \right |_{2} =\left ( \sum_{i,j=1}^{n} a_{ij}^{2} \right )^\frac{1}{2}$ for $A\in \mathbb{R}^{n\times n}$. Show that this defines a matrix norm. I remember i've ...
0
votes
1answer
59 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 $f(x+y) \leq f(x) + f(y)$?
0
votes
1answer
38 views

A question on minimizing $\| . \|_2^2$ vs $\| . \|_2$

Suppose we are in $\mathbb{R}^n$ Is the problem of $d(x,Y) = \inf\{ \| x - y\|^2 : y \in Y\}$ equivalent to $d(x,Y) = \inf\{ \| x - y\| : y \in Y\}$ Pardon me, let us keep it simple and just stick ...
0
votes
2answers
849 views

Show these two norms are not equivalent?

I have the following two norms on $C[a,b]$ : $$||x||_1= \int_a^b |x(t)|dt$$ $$||x||_\infty = sup_{t \in [a,b]}|x(t)|$$ $\forall x \in C[a,b]$. I need to prove that these are not equivalent. They are ...
0
votes
3answers
74 views

Compute the norm of matrix

Let $M$ be $n\times n$ matrix, consisting entirely of 1's. Show, that $\|M\|_{op}=\sup_{x\in C^n}|Mx|=n$.
0
votes
1answer
43 views

Integral like a norm instead of sum

A Rienmann integral is defined as: $\int_a^b f(x)\ dx=\displaystyle\lim_{n\rightarrow\infty}\displaystyle\frac{1}{n}\displaystyle\sum_{i=0}^{n}f\left(\displaystyle\frac{b-a}{n}\cdot i+a\right)$ I ...
0
votes
2answers
311 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.
0
votes
2answers
37 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). ...
0
votes
2answers
296 views

Variation of reverse triangle inequality

I know from the reverse triangle inequality that for $x,y \in \mathbb{R}^n$ the following holds: $ \vert x \vert - \vert y \vert \leq \vert x -y \vert $ but does also this one hold? $ \vert x ...
0
votes
1answer
276 views

norm induced by inner product and triangle inequality

Let $\langle\cdot,\cdot\rangle$ be a scalar product on a space $X$, and let $\lVert \cdot\rVert$ denote the norm induced by this scalar product. I need to show that for $x,y\in X$, $\lVert ...
0
votes
1answer
88 views

Analysis.. Convergence of sequence

I really struggle with understanding convergence and have the following questions.. Determine whether the following sequences converge and if so, give the limit: $x_n = ...
0
votes
1answer
489 views

Ky Fan Norm Question

How can one simply see that Ky Fan $k$-norm satisfies the triangle inequality? (The Ky Fan $k$-norm of a matrix is the sum of the $k$ largest singular values of the matrix) Thanks.
0
votes
1answer
34 views

Upper bound of Frobenius norm of product of matrices.

I'm trying to prove that $||AB||_F\leq||A||_2||B||_F$. As far as I know it isn't a hard problem but I was stuck. Any ideas?
0
votes
1answer
11 views

Triangle Inequality for SPD Matrix Norm

We define a symmetric, positive-definite matrix $A$ to be one such that $A = A^T$ and for $x \neq 0$, $x^TAx > 0$. If we have a norm $\|x\|_A = \sqrt{x^TAx}$, how can we show the triangle ...
0
votes
2answers
29 views

Proving a norm on the space of differentiable functions

I consider the space $C^1[a, b]$ of (complex) functions that are at least once differentiable on $[a, b]$. I want to show that $$||f||_{C^1} := ||f||_\infty + ||f'||_\infty$$ defines a norm on ...
0
votes
1answer
43 views

Minimize the norm of $w$.

Why is minimizing the norm of $w$ equivalent to minimizing $\frac{1}{2} \cdot |w|^2$? I have tried to derive the norm but the result is the following $$\frac{1}{2 \cdot |w|}$$
0
votes
1answer
27 views

Prob. 3, Sec. 2.8 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $C[-1,1]$ denote the normed space of all (real or complex-valued) functions defined and continuous on the closed interval $[-1,1]$ on the real line, with the norm given by $$\Vert x ...
0
votes
1answer
32 views

What is the norm of a complex vector?

I have two arrays $a$ and $b$ containing complex values. Now I one of my target operations is the following: $$||a-b||$$ The result should be a single real number. Now I am a bit confused how to apply ...
0
votes
2answers
41 views

Frobenius Norm to L2 norm Problem

Here is the problem: if $v^1$, $v^2$, ..., $v^d$ is an orthonormal basis in $\mathbb{R}^d$, then show that $$ ||A - A\sum_{i = 1}^k v^i(v^i)^T ||^2_F = \sum_{i = k+1}^d||Av^i ||_2^2. $$ I am having ...
0
votes
2answers
52 views

Requirement on Norm for units in Cyclotomic Fields

Consider $\zeta_p$ a primitive $p^{th}$ root of unity. Prove that $\alpha \in \mathbb Z[\zeta_p]$ is a unit iff $N(\alpha)=\pm1$. I'm not even sure where to start with this. I know that ...
0
votes
2answers
45 views

Bound on Symmetric Matrices

Let $A=(a_{ij})$ be a matrix with real entries, $1 \leq i, j, \leq n$. Let $A^{T}=(a_{ij}^{T})$ be the transposed matrix, that is $a_{ij}^{T}=a_{ji}$. Suppose that $a_{ij}=a_{ji}$, namely $A$ is a ...
0
votes
1answer
47 views

How do you express the Frobenius norm of a Matrix as the squared norm of its singular values?

Let the Frobenius norm of an m by n ($m \times n$) matrix M be: $$|| M ||_{F} = \sqrt{\sum_{i,j} M^2_{i,j}}$$ I was told that it can be proved that, if M can be expressed as follows (which we can ...
0
votes
1answer
37 views

Why is the euclidean norm not differetiable at $0$?

I denote $N(x)$ as the norm-function, although in the denominator it stays $\|x\|$. $$\lim_{x\to 0} \frac{N(x)-N(0)}{\|x\|} = \lim_{x\to 0} \frac{N(x) - 0}{\|x\|} = \lim_{x\to 0} 1 = 1 \ne 0$$ 1) ...
0
votes
1answer
46 views

Verifying that the Sobolev space is a Banach Space

In PDE Evans 2nd edition (pages 262-263), I am trying to understand a proof of a theorem, which states: THEOREM 2 (Sobolev spaces as function spaces). For each $k=1,\ldots$ and $1\le p\le\infty$, ...
0
votes
1answer
52 views

What does it mean to write $|||x|||$ rather than $||x||$?

I am familiar with the notation $||x||$ meaning some norm of $x$. I have just come across the notation $|||x|||$ (in a text that also uses the former for norms). What is the difference?
0
votes
1answer
60 views

An inequality about operator norm

Let $H$ be a Hilbert space and $T\in B(H)$, with $T_{i}\rightarrow T$ in strong operator topology. Then can we prove that $\liminf_{i\rightarrow \infty}||T_{i}||\geq ||T||$ ?
0
votes
1answer
50 views

Norms for which every subset of closed unit ball containing the open unit ball is convex

It can be shown without much difficulty that any Euclidean norms satisfies the following condition :$$(P) \quad B \subset X \subset B' \Rightarrow X \, \text{is convex}$$ where $B=\{x \in E / \|x\| ...
0
votes
1answer
62 views

Find the norm of the operator $T:\ell^2 \to \ell^2$ defined by $Tx := (x_1, x_1+x_2, x_3, x_3+x_4, x_5, x_5+x_6, \ldots)$

Let $T : \ell^2 \to \ell^2$ (involving complex numbers) be defined by $$ Tx := (x_1, x_1+x_2, x_3, x_3+x_4, x_5, x_5+x_6, \ldots). $$ What is $\|T\|$? Essentially I've tried : To find $M \geq 0$ ...
0
votes
1answer
98 views

Lipschitz continuity and gradient of a real-valued function on a normed space

The gradient of a function $f:E\to\mathbb{R}$ is Lipschitz continuous with parameter $L > 0$ iff $$\|\nabla f(x) - \nabla f(y)\|^* \le L\|x-y\| \quad \forall x,y\in E.$$ I have two questions: ...
0
votes
2answers
41 views

Express Norm Using Inner Product

I'd like to know whether there's a way to express a norm using inner product, for example , is there any inner product we may use that is equal to $(||Ax-b||_2)^2$ ? Thanks in advance.
0
votes
1answer
62 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 ...
0
votes
2answers
28 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
votes
1answer
46 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\ ...
0
votes
1answer
284 views

What minimizes the Chebyshev Distance?

For an arbitrary number of dimensions, I know that the mean minimizes the distance using the L2 norm and that the geometric median minimizes the distance function using the L1 norm (though I have yet ...
0
votes
1answer
87 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
votes
1answer
57 views

Check that: it is a norm

My question is: Let $f \in C^{1} [0,1]$ & let $f'$ denote its derivative. Define: $|| f ||_{1} = ( \int_{0}^{1} (|f(t)|^{2} + |f'(t)|^{2})dt)^{\frac{1}{2}}.$We are to show that: $||f||_{1}$ ...
0
votes
2answers
47 views

Show that the set of points that are nearer $a$ than $b$ with respect to $\lVert \cdot \rVert_2$ is convex

I am trying to show the above statement: Show that the set of points that are nearer $a$ than $b$ in the sense of Euclidean $\lVert\cdot\rVert_2$ are convex. My attempt This follows from the ...
0
votes
1answer
597 views

Norm of Matrix transpose

I have a problem below: Let $\|\cdot\|$ denotes the norm matrix \begin{equation} \|A\|=\max \frac {\|Ax\|}{\|x\|}, \end{equation} for every $A$. Now suppose that $H: \mathbb{R}^k \rightarrow ...
0
votes
1answer
65 views

A norm inequality $(AB+BA') \leq |A+A'|*B \leq 2|A|*B$

Suppose that A and B are two $n \times n$ matrices. If $AB+BA'$ and $B$ are both positive definite symmetric matrices, is it true to conclude that $AB+BA' \leq $|A+A'|*B$ \leq 2|A|*B$? $A<B$ we ...
0
votes
2answers
43 views

If $||f-g|| < ||f^{-1}||^{-1}$, then $f$ is isomorphism implies $g$ is also isomorphism

Let $E$, $F$ be Banach space, $f,g \in L(E,F)$ and $f$ is isomorphism. Prove that if $||f-g|| < ||f^{-1}||^{-1}$, then $g$ is isomorphism. Hi everybody. I got stuck on this problem and can't ...
0
votes
1answer
62 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 ...
0
votes
2answers
241 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 ...
0
votes
1answer
54 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. ...
0
votes
2answers
127 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 ...
0
votes
1answer
45 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
1answer
51 views

In $C(\mathbb{R})$, the vector space over $\mathbb{R}$ of continuous functions from $\mathbb{R}$ to $\mathbb{R}$

In $C(\mathbb{R})$, the vector space over $\mathbb{R}$ of continuous functions from $\mathbb{R}$ to $\mathbb{R}$, let $W$ = { f ∈ $C(\mathbb{R})$ | $∫_{-∞}^∞$ | f(x) | $dx$ < ∞} where the ...