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
55 views

Norm to evaluate precision

I have two vectors x and y with their respective n coordinates/components and this norm between them: $$ norm = \sqrt{\frac{\displaystyle\sum_{i=1}^{n} (x_{i}-y_{i})^2}{\displaystyle\sum_{i=1}^{n} ...
3
votes
1answer
449 views

How to show the unit ball of the dual norm is also polytope?

Assuming the norm's unit ball is a convex polytope. How can one show that the unit ball of the dual's norm is convex polytope and/or polytope ?
14
votes
1answer
890 views

On the norm of a quotient of a Banach space.

Let $E$ be a Banach space and $F$ a closed subspace. It is well known that the quotient space $E/F$ is also a Banach space with respect to the norm $$ \left\Vert x+F\right\Vert_{E/F}=\inf\{\left\Vert ...
2
votes
0answers
155 views

Step functions dense in Integrable functions with respect to $L_2$

Let $I$ be a bounded interval. Prove that $\{\text{step functions }I \to C\}$ is dense in $\{\text{integrable functions }I \to C\}$ (Riemann Integrable) with respect to $\|.\|_2$ ($L_2$ norm)
1
vote
1answer
75 views

On the convexity of the element-wise norm 1 of a pseudoinverse

Let us define $\|A\|_1$ the element wise norm 1 of a matrix $A \in \mathbb{R}^{n \times m}$ as $$ \|A\|_1= \sum_{i,j} |A_{i,j}|. $$ Obviously, this function is convex over $\mathbb{R}^{n \times m}$. ...
3
votes
0answers
73 views

Exercise from textbook about norm

The below diagram included a relevant page and the question 4 of the exercise which i am not sure how to do. Any hint or solution are welcome Attempts: a) I have done it b) i have tried to show that ...
0
votes
1answer
34 views

A limit superior question in the context of the Neumann series

I'm trying to understand a step in the proof that the Neumann series converges: Let $X$ be a Banach space and $T\in L(X)$ (the space of bounded, i.e. continuous linear maps $X\to X$). It is known ...
0
votes
1answer
60 views

Approximating asymmetric matrix's Ky Fan norm

Given an N-by-N asymmetric matrix $M$ Is there any theory about approximating $M$'s Ky Fan k-norm $|| M ||_k$ using $\frac {(M+M^T)}{2}$'s Ky Fan k-norm $|| \frac {(M+M^T)}{2} ||_k$? UPDATE: ...
1
vote
2answers
143 views

minimizing a norm and a linear function

Let $y,\lambda\in\mathbb{R}^n$. I want to minimize the following with respect to $y$. $$ f(y)=||y|| + \lambda^Ty $$ where $||y||$ is the Euclidean norm. I first take the derivative of the function and ...
1
vote
2answers
223 views

Norms of idempotent matrices

So if I have a random idempotent matrix (a matrix that when squared equals itself) how do I go about calculating its 2-norm? I know that a idempotent matrix has eigenvalues of only 0 or 1 and I know ...
3
votes
2answers
223 views

Matrix proof using norms

I have a linear algebra question I need help with. Let $A$ be an $m\times m$ matrix with $\|A\|_2 < 1$ where $\|A\|_2$ is the $2$-norm of $A$. Show that $I - A$ is invertible where $I$ is the ...
2
votes
1answer
230 views

Need help understanding matrix norm notation

I've been trying to understand matrix norms (full disclosure: school assignment, not looking for answers, just clarity!), and how they follow from vector norms - been awhile since I did much linear ...
2
votes
1answer
98 views

What are the maximumnorm and supremumnorm of a vector when having a basis?

I have a perhaps stupid question. When having a finite-dimensional Vectorspace $X$ (f.e. n-dimensional) and when knowing a basis $V=\left\{v_1,...,v_n\right\}$ of it, so any $x\in X$ can be written as ...
1
vote
1answer
128 views

Matrix Trace representation?

For a real, symmetric matrix $A$ and a real, rectangular matrix $X$, am looking for a matrix trace based representation of this simple linear algebraic expression $\sum_{i} A_{ii} ...
2
votes
2answers
203 views

Derive steepest descent vector subject to norm constraint

I am currently working through an old textbook Practical Optimization by Gill, Murray and Wright (c 1982) who make some derivations which seem correct, but I am unable to duplicate. In the equations ...
2
votes
1answer
56 views

To show the inequality $\|A\|\geq\max\{\|u_1\|,\ldots,\|u_q\|,\|\vec{v_1}\|,\ldots,\|\vec{v_q}\|\}$

Let $A\in$ $\mathbb{C}^{p\times q}$ with column $u_1,\ldots,u_q$ and rows $\vec{v_1},\ldots,\vec{v_p}$. show that $$\|A\|\geq\max\{\|u_1\|,\ldots,\|u_q\|,\|\vec{v_1}\|,\ldots,\|\vec{v_q}\|\}$$ and ...
3
votes
1answer
354 views

Proof of a norm

Prove that: $ \|\cdot\|_1:\mathbb{R}^n \rightarrow \mathbb{R};\vec{x} \mapsto \sum_{j=1}^n |x_j| $ is a norm defined on the vector space $\mathbb{R}^n$. 1) Zero vector: $\sum_{j=1}^n |x_j| = |x|_1 ...
0
votes
1answer
84 views

Norm of normal Operator A

I just found the following equality $||A||=sup_{\lambda\in\sigma(A)} |\lambda|$ My question: What exactly means $\sigma(A)$ and why this is true ? I always thouht the only way to get the ...
0
votes
1answer
119 views

Hilbert norm and Euclidean distance

For real matrix $X$ where $d_{i,j}^2(X)$ indicates the euclidean distance squared between the rows $i,j$ of $X$, if $d_{i,j}^2(X)=||f(X_i.)-f(X_j.)||_H$ then what would the function $f(.)$ be? Is ...
5
votes
2answers
312 views

Cauchy-Schwarz for metrics with arbitrary signatures

When the norm of a vector is always greater than or equal to zero, the Cauchy-Schwarz inequality holds, but what if we look at a metric with an arbitrary signature? Then the inner product of a vector ...
1
vote
1answer
186 views

p norm Matrix relationship

I am trying to show that $\Vert A \Vert_\infty \leq \sqrt{n}\Vert A \Vert_2$ given that $A \in \mathbb{R}^{m \times n}$, $\Vert A \Vert_\infty = \max \limits_{1\leq i \leq n} \sum\limits_{j=1}^n ...
2
votes
2answers
181 views

equivalence of norms and direct sum

Let $(X,\|\cdot\|_X) $ be an infinite dimensional Banach space. Suppose that you can write $X=V\oplus W$. Write $x=v+w$ and define $(V\oplus W,\|x\|_\oplus :=\|v\|_X+\|w\|_X)$. Show that ...
1
vote
1answer
73 views

Norm of Polynom

I'm trying to understand the following equation: $$\langle p,q\rangle_p=\int_{-1}^1p(x)q(x)dx\\ \text{Basis: }\{p_1,p_2\}\\ p_1:=2x,\quad p_2:=x-1$$ Norm: $$p_1:q_1=\frac{p_1}{\|p_1\|_p}$$ ...
5
votes
2answers
270 views

What is the reason norm properties are defined the way they are?

Suppose we have a complex vector space $V$. A norm is a function $f : V \rightarrow \mathbb{R}$ which satisfies (i) $f(x) \ge 0$ for all $x \in V$ (Positivity - Non-Negativity) (ii) $f(x + y) \le ...
4
votes
1answer
382 views

Poincaré's lemma with norm in $H_{0}^{1}$

I wonder why $\| u\|_{H_0^1} = \int_{\Omega} |Du|^2$ for u in $H_0^1(\Omega)$, with $\Omega = (-1,1)$? I might be wrong but isn't $\| u\|_{H^1} = \| u\|_{L^2} + \| Du\|_{L^2}$? How come that $\| ...
1
vote
1answer
192 views

Equivalence of Norms Defined on a Cartesian Product

While studying some notes on normed vector spaces, I have come upon the proof that addition $+:V \times V\to V$ of vectors in a normed vector space $V$ is a continuous operation. The proof of this ...
1
vote
1answer
202 views

Dual norm and distance

Let $Z$ be a subspace of a normed linear space $X$ and $x\in X$ has distance $d=\inf\{||z-y||:z\in Z\}$ to $Z$. I would like to find a function $f\in X^*$ that satifies $||f||\le1$, $f(x)=d$ and ...
0
votes
1answer
141 views

Inequality for singular value for differences of matrices (upper bound)

Does anybody know the inequality of singular value for differences of matrices, i.e. $\sigma_{max}\left(\begin{array}{c} A-B\end{array}\right)\leq??$ in term of $\sigma_{max}\left(\begin{array}{c} ...
5
votes
2answers
141 views

Calculate $\left\Vert \begin{bmatrix}1 & 2\\ 2 & 4 \end{bmatrix} \right\Vert$

With $$\left\Vert A \right\Vert=\max_{\mathbf{x}\ne 0}\frac{\left\Vert A\mathbf{x}\right\Vert }{\left\Vert \mathbf{x}\right\Vert }$$ and $$A=\begin{bmatrix}1 & 2\\ 2 & 4 \end{bmatrix} $$ ...
3
votes
1answer
141 views

Absolute norms and 1-unconditional sums

Absolute norm Let $X$ and $Y$ be Banach spaces. Let $Z=X\times Y$ a norm $\|\cdot\|_N$ on $Z$ is called absolute if there is a function $N\colon R^2\rightarrow R$ such that $$ \|(x,y)\|_N=N((\|x\|, ...
0
votes
1answer
224 views

Is there any main method for finding norm of function in $L_1$ space?

Is there any main method for finding norm of function in $L_1$ space? For example : $f(x)$ = $\sin x$ in space $L_1[-\pi,\pi]$
1
vote
1answer
57 views

Single norm criterion

Let $E$ be a metrizable locally convex space whose topology is defined by an increasing sequence $\{p_n\}$ of seminorms. Show that the topology of $E$ can be defined by a single norm iff there ...
2
votes
1answer
306 views

Upper and lower bounds of a ratio involving vector norms

I'm working on a signal processing problem and need to analyze the following expression $$ G = \frac{n}{\sum\limits_{i=1}^n |w_i|} \frac{ \sum\limits_{i=1}^n g_i w_i^2}{\sum\limits_{i=1}^n g_i |w_i|} ...
1
vote
2answers
104 views

Matrix norm less than $1$ iteration

Is the following true always for a matrix norm $$\lVert AB\rVert \leqslant \lVert A\rVert \cdot \lVert B\rVert \text{ ?}$$ Related to this given $r$ is positive constant, $H$ is symmetric positive ...
2
votes
1answer
48 views

Prove that absorbing sets induce quasinorms

I have been trying to prove this for so long that I pose this to Math.SE: Let D be a bounded (in regards to, for example, the maximum norm), absorbing subset of a finite-dimensional vector space V ...
1
vote
0answers
208 views

Why is the subdifferential of norm of a matrix ||A|| defined like this?

I read in a paper called "Characterization of the subdifferential of some matrix norms" that it defines the subdifferential of the matrix norm like this: $$\partial ||A||=\{G \in R^{m\times n} : ...
1
vote
1answer
109 views

norm of a matrix ( which norm have to use ?)

I need to find the norm of the matrix $$ A=\left( \begin{array}{cc} e^{-x} \cos( \sin x) & e^{-x} \sin ( \sin x) \\ -e^{-x} \sin ( \sin x) & e^{-x} \cos (\sin x) \end{array} \right) $$ Here ...
3
votes
0answers
91 views

Is $\rho(f) := \mu(|f| \geq \epsilon), \epsilon >0$ some kind of generalized norm?

Let $\rho(f) := \mu(|f| \geq \epsilon)$ where $\epsilon >0$ is a constant and $\mu$ is a measure and $f$ is a measurable function. Then $f\equiv 0 \rightarrow \rho(f) = 0$ when $|a| \neq 0$, ...
1
vote
1answer
43 views

Question about norms

Denote supnorm by $|x| = \max\{|x_1|,...,|x_n|\}$ where $x \in \mathbb{R}^n$. How can we show that this norm and Euclidean norm satisfies the following inequality? $$ |x| \leq ||x|| \leq ...
1
vote
1answer
102 views

Small question regarding norms and Holder conjugates.

I'm trying show that if $p,q$ are Holder Conjugates then: $$\forall\, a\in\mathbb{R}^{n}:\,\Vert a\Vert_{q}=\max_{x\in\mathbb{R}^{n},\,\Vert x\Vert_{p}=1}\left<a,x\right>$$ Where ...
3
votes
1answer
316 views

Matrix norm inequality implying eigenvector norm inequality

For a matrix $A$ let $\|A\|$ be the norm given by $\|A\|=\sup_{v \neq 0}\frac{\|Av\|}{\|v\|}$ where $\|v\|$ is the Euclidian norm on the vector $v$. Suppose we have matrices $M$ and $S$ with leading ...
3
votes
4answers
266 views

The difference between $L_1$ and $L_2$ norm?

I have been trying to understand what is the difference between $L_1$ and $L_2$ norm and cant figure it out. In this webpage I got a clear understanding of why we would use $L_1$ norm (scroll down ...
4
votes
5answers
75 views

Norm of vector in $\mathbb{R}^3$ with multiple

If we have a vector in $\mathbb{R}^3$ (or any Euclidian space I suppose), say $v = (-3,-6,-9)$, then: May I always "factor" out a constant from a vector, as in this example like $(-3,-6,-9) = ...
1
vote
1answer
104 views

Matrix norm relationship

Suppose we have two matrices, A and B, and $\left\Vert A\right\Vert _{F}\geq\left\Vert B\right\Vert _{F}$ where $\left\Vert .\right\Vert _{F}$ denotes Frobenius norm. Does it imply $\left\Vert ...
3
votes
2answers
131 views

Linear Algebra munkres analysis on manifolds question.

If $A$ is an $n$ by $m$ matrix and $B$ is an $m$ by $p$ matrix, then $$ |AB| \leq m|A||B|$$ where $|A| = \max\{|a_{ij}| : i = 1,\ldots,n \text{ and} j = 1,\ldots,m\}$ Attempt: $ |AB| = \max\{| ...
5
votes
1answer
317 views

Norm in a dual space

If $f \in X^*$, with $X^*$ the dual space consisting of all linear bounded functionals on a linear normed space $X$. With the norm defined as $||f||_{X^{*}} = \sup_{||x|| \leqslant 1} |f(x)|$. Why ...
8
votes
3answers
333 views

Attaining the norm of an ideal in a number field by the norm of an element

Let $K$ be a number field of degree $n$ and $\mathfrak{a}$ be an ideal in its ring of integers $\mathcal{O}_K$. We can consider: The norm $N(\mathfrak{a})$ of $\mathfrak{a}$. The norms $N(x)$ of the ...
2
votes
2answers
52 views

Why the max can be found only with normalized vectors?

Can someone explain why the term in the first {} equals the second term in the {} :
2
votes
2answers
576 views

Proof of Clarkson's Inequality

Trying to find a proof for Clarkson's inequality, which states that if $2 \leq p < \infty$, then for any $f, g \in L^p$, we have that $$\left|\left|f+g\right|\right|_p^p + ...
1
vote
0answers
106 views

How to represent and handle eigenvalue constraint in optimization

For example consider the problem $\min_X f(X)$ s.t. $\lambda_i(X+A)=\lambda_i(B)$ for $i \in {1,...,N}$ where $A$ and $B$ are full rank N by N matrix, $\lambda_i(X)$ is the i-th eigenvalue of $X$ ...