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

1
vote
1answer
30 views

Quadratic form $x^T A x $ is less than $||A|| x^T x$?

Does the following hold? $$x^T A x \leq ||A|| x^T x$$ $A$ is a symmetric positive-definite real-valued matrix, and $x$ is a real-valued vector. Norm is Euclidean.
1
vote
1answer
48 views

What is the norm of this linear operator?

I need to show that $\|A(g_n)\|_1=b-a-\frac1n$ where $A:L^1([a,b])\to L^1([a,b])$ is given by $A(f)(x) = \displaystyle\int_a^xf(t)~dt$, and $g_n:[a,b]\to\mathbb R$ is given by $$g_n(t)= \begin{cases}n ...
1
vote
1answer
38 views

Prove $\| \cdot \|_{\infty}$ is well-defined on $\ell ^1$

$1 \leq p < q \leq \infty$ $p=1$ and $q= \infty$ Let $(a_k)_{k \geq 1} \in \ell^1$, then $$\sum_{k \geq 1}|a_k|^1< \infty$$ (*) which means $\sup_{k \geq 1 } |a_k|^1 < \infty$. Hence it is ...
1
vote
1answer
63 views

Sequence of partial sums converge

Let $(X,\|.\|)$ be a Banach space and a sequence $(x_n) \subseteq X$ be such that the series $\sum_{n=1}^{\infty} \|x_n\|$ converges in $\mathbb R$. Prove that the sequence $S_n=x_1+...+x_n$ converges ...
1
vote
1answer
45 views

Proving these norms are not equivalent

$X$ is the vector space containting all polynomials with real coefficients. For every $P \in X$, define $N_1(P)= \sup _ {t \in [0,1]} |P(t)|$ and $N(P)=N_1(P)+|P'(1)|$. Prove that $N$ is not ...
1
vote
1answer
48 views

Are some of the axioms of a norm of a vector space unnecessary?

I have a homework problem where my task is to find out if some of the axioms of a norm of a vector space are unnecessary, meaning they can be derived from other axioms (I presume from the problem ...
1
vote
1answer
91 views

Two (equivalent ?) norms on Hilbert space

Let $H$ be a vector space equipped with two inner products $\langle \cdot,\cdot\rangle_1, \ \langle \cdot,\cdot\rangle_2$, s.th. $(H,\langle \cdot,\cdot\rangle_j)$ is a Hilbert space for $j=1,2$. ...
1
vote
3answers
33 views

Prove that $\| A \| = \lvert y \rvert$

For all $A \in L(\Bbb R^n, \Bbb R)$ there is a unique $y \in \Bbb R^n$ such that $A\textbf{x} = \textbf{x} \cdot \textbf{y}$. Prove that $\| A \| = \lvert \textbf{y} \rvert$. Hint: Cauchy-Schwarz ...
1
vote
1answer
48 views

$A$ is normal, and $B$ is Hermitian, why does $\left\| \rm{AB} \right\|{\rm{ = }}\left\| \rm{BA} \right\|$?

Let $\left\| . \right\|$ be a unitarily invariant norm on $M_n$. If $A, B ∈ M_n$, $A$ is normal, and $B$ is Hermitian, why does $\left\| \rm{AB} \right\|{\rm{ = }}\left\| \rm{BA} \right\|$?
1
vote
1answer
49 views

$|x+y|=|y+x|$ in a normed group

A normed group $(X,+,|\cdot|)$ is a set $X$ equipped with a group operation $+$ and a function $|\cdot|:X\to\Bbb R$ called a norm such that $|x|=0\iff x=0$ $|x-y|\le|x|+|y|$. From these ...
1
vote
1answer
66 views

Vector norm inequality [duplicate]

Is it true that for vectors $x$ and $y$ in $\mathbb R^n$ $|\Vert x\Vert -\Vert y\Vert| \ge \Vert x-y\Vert $? Can I simply use the triangle inequality $\Vert x\Vert +\Vert y\Vert \ge \Vert x+y\Vert $ ...
1
vote
1answer
39 views

If $||A^N|| < 1$ then is ||A||<1?

following a previous question in functional analysis I asked Let $X$ be a banach space and $A$ is a bounded linear operator on $X$, and there exists some natural $N$ for which $ ||A^N|| < 1 $. ...
1
vote
1answer
46 views

Can someone explain why $\| \cdot \|_p, p = \frac{1}{2}$ isn't a norm?

$\| \cdot \|_p, p = \frac{1}{2}$ or "half norm" is not a norm What is a quick way to verify that it is indeed not a norm?
1
vote
1answer
79 views

Proving $\lVert x \rVert_3$ is smaller than euclidean norm

Is there a simple way to show that $\lVert x \rVert_3\leqslant \lVert x \rVert_2$ for vectors in $\mathbb{R}^2$? I've used the result in another problem but can't figure out where to get it from.
1
vote
2answers
48 views

Proving an induced operator norm equality:

The induced matrix norm is defined by $$||A|| = \sup_{x \ne 0} \frac {||Ax||}{||x||} $$ Show that $$||A|| = \sup_{||x||=1} ||Ax||$$ A is only assumed to be square -- not anything more, e.g., not ...
1
vote
1answer
86 views

Why is the Euclidian norm used to measure complex numbers?

Why is the Euclidian norm used to measure complex numbers? The complex numbers are numbers (or more precisely, pairs of numbers), and I can't see why are they essentially connected to the Euclidian ...
1
vote
1answer
36 views

Is $\left\|A^TA(x-y)\right\| = \left\|A^TA\right\|\times \left\|x-y\right\|$ correct? $A \in \mathbb{R}^{n \times n}$

In the derivation of following, I meet a dumb problem: Note: 1. $\left\|\: \cdot \,\right\|$ is the $l_2$ norm. 2. $A \in \mathbb{R}^{n \times n} $ 3. $x,y \in \mathbb{R}^{n}$ $$\frac{\left\|A^...
1
vote
1answer
215 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?
1
vote
2answers
131 views

Prove that there is an inner product on $\mathbb{R}^2$ such that the associated norm is given by…

Suppose $p>0$. Prove that there is an inner product on $\mathbb{R}^2$ such that the associated norm is given by $\|(x,y)\|=(x^p+y^p)^\frac{1}{p}$ for all $(x,y)\in\mathbb{R}^2$ if and only if $p=2$....
1
vote
4answers
334 views

Euclidean norm second derivative

I really need Your help. I need to prove that Euclidean norm is strictly convex. I know that a function is strictly convex if $f ''(x)>0$. Can I use it for Euclidean norm and how? $||x||''=\frac{||...
1
vote
1answer
124 views

Linear and nonlinear operator on normed space and its properties

My first question is : We know every bounded linear operator is continuous operator , and every continuous linear operator is bounded operator , in other words the continuousness and boundedness are ...
1
vote
2answers
31 views

Norm controls the components of a matrix- proof?

For certain norms (such as the Frobenius norm) it is clear that the norm provides component-wise control - each component is at most as large as the norm in magnitude. How do we establish this for ...
1
vote
1answer
53 views

Bounding a Symmetric Matrix

Consider the following $n \times n$ matrix $A$, which has 1's on the superdiagonal and subdiagonal and 0's elsewhere, i.e. $$\begin{pmatrix} 0 & 1 & 0 & \cdots & \cdots & \cdots &...
1
vote
1answer
872 views

Show that the dual norm of spectral norm is Nuclear norm.

Could someone help to understand that the dual norm of spectral norm is Nuclear Norm ? We can focus on the real field. Given a matrix $X \in \mathbb{R}^{mn},$ then the spectral norm is defined by: ...
1
vote
2answers
82 views

How do I find the norm of a matrix?

I have the following matrix below and I would like to find the norm of the matrix. I learned that the norm of a matrix is the square root of the maximum eigenvalue multiplied by the transpose of the ...
1
vote
2answers
71 views

A condition equivalent to orthogonality

Prove that in any inner product space: $x$ and $y$ are orthogonal if and only if $||x+\alpha y||\ge ||x||$ for every scalar $\alpha$.
1
vote
1answer
162 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 $1\le p\le\infty$, ...
1
vote
1answer
43 views

Unit ball for a special norm

What does the unit sphere for the norm on $\mathbb{R}^2$, $\displaystyle N(x,y)\rightarrow\sup_{t\in\mathbb{R}}\frac{|x+ty|}{t^2+t+1}$, look like ? My approach was to consider $y=ax$ so as to get $...
1
vote
1answer
39 views

Weighted Norm Minimization

I have a minimization problem of the form $min (w_1 \|x\|+w_2\|y\|)$ subject to constraints $A_1x=b_1$, $A_2y=b_2$, $0< l_1 \leq x \leq u_1$ and $0< l_2 \leq y \leq u_2$ where $l_1,l_2,u_1,u_2\...
1
vote
1answer
61 views

Is every regular polygon the unit ball for some norm?

For every regular polygon, is there a norm such that the polygon is it's unit ball centered on 0?
1
vote
1answer
72 views

Some Norms on $V=P_n(\mathbb{R})$?

$V=P_n(\mathbb{R})$ be the vector space of all polynomials with degree $\le n$. I need to know Which of the followings are norms on $V.$ $\forall p\in V$ 1.$\|p\|^2=|p(1)|^2+\dots+|p(n+1)|^2$ 2.$\|...
1
vote
2answers
39 views

Projection function preserving distance

Given $n>1$. I wis to construct a function $f:\mathbb{R}^n\mapsto\mathbb R$ such that $$\|X_1-X_2\|\le\|Y_1-Y_2\|\implies|f(X_1)-f(X_2)|\le |f(Y_1)-f(Y_2)|$$ for all $X_1,X_2,Y_1,Y_2\in\mathbb{R},$ ...
1
vote
2answers
85 views

Is a minimum a minimum in every norm? (for finite-dimensional vector spaces)

I have a question regarding the equivalence of the norms in finite-dimensional vector spaces. Basically the question is: if $\hat{x}$ is some minimum-norm solution in a subspace $\mathcal{K}$ under ...
1
vote
1answer
191 views

The supremum norm is submultiplicative

Is the following proof correct: Let $X$ be compact a compact Hausdorff space and $C(X)$ the continuous functions $f: X \to \mathbb{C}$ on X. We can equip $C(X)$ with the (edit: sorry, semi-)norm $\...
1
vote
1answer
54 views

Showing that a map $x \to \|x\|$ is continuous?

I am given this: Consider a real Banach space $X$ with norm $\|*\|$. 1) Show that the map $x\to \|x\|$ from $X$ to $\mathbb{R}$ is continuous. Is it uniformly continuous? 2) Show that the maps $(x,...
1
vote
1answer
77 views

Why is $L_0$ norm not convex? [closed]

I have this confusion in understanding the convexity of the $L_0$ norm. Why is $L_0$ norm not convex?
1
vote
1answer
88 views

Find the norm of the operator $A:L_2[0,2] \rightarrow L_2[0,2]$ defined by $(Ax)(t) = t \, \operatorname{sgn}(t-1) x(t)$

I have operator: $\boldsymbol{L}_2[0,2] \to \boldsymbol{L}_2[0,2], ( Ax)( t ) = \boldsymbol{t} \operatorname{sgn}(t-1)x(t)$ I need to find operator norm or say that operator isn't bounded. ...
1
vote
1answer
107 views

Are the norms $\Vert f\Vert_1=\Vert f\Vert _\infty+\Vert f'\Vert _\infty$ and $\Vert f\Vert _2=\vert f(a)\vert +\Vert f'\Vert _\infty$ equivalent?

We have the norms $\Vert f\Vert_1=\Vert f\Vert _\infty+\Vert f'\Vert _\infty$ and $\Vert f\Vert _2=\vert f(a)\vert +\Vert f'\Vert _\infty$ where $f\in C^1[a,b]$. Are they equivalent and how shoud I ...
1
vote
2answers
1k views

Consistency of matrix norm: $||Ax||_2 \leq ||A||_{Frobenius}||x||_2$

I'm trying to show that $||Ax||_2 \leq ||A||_{F}||x||_2$ where $A$ is an n by n matrix, $x\in \mathbb R^n$, $||x||_2$ is the euclidean norm, and $||A||_F$ is the frobenius norm. I actually wrote ...
1
vote
1answer
70 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
78 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 ...
1
vote
1answer
319 views

L1-norm minimization

This is undoubtedly a trivial question but might as well ask: Why is the L1 norm minimization a heuristic for finding the sparsest vector? What I mean is that if the L1 norm sums the elements of a ...
1
vote
1answer
121 views

Matrix norm equivalence

If we define $ \|A\| = \max \{|A\cdot \mathbf{t}|:|\mathbf{t}|\leq 1\}.$ is it the same as defining it as $\max \{|A\cdot \mathbf{t}|:|\mathbf{t}|= 1\}$ ? If so, why? The book I'm following uses the ...
1
vote
1answer
55 views

Inequality of scalar-product and norm

Why does the following inequality hold, given $A$ is symmetric and $\lambda_{\min} (A)$ is the smallest Eigenvalue of $A$? $$v^\top A v \ge \lambda_{\min} (A) \; ||v||^2$$
1
vote
2answers
3k views

Frobenius Norm Triangle Inequality

How can I go about proving the triangle inequality holds for the Frobenius norm? I worked through $\|A+B\|_F \le \|A\|_F + \|B\|_F$ and was not able to make it work =/.
1
vote
1answer
1k views

Frobenius Norm Properties

Say $v\in {\mathbb{R}^{l}}$ (a column vector) and $A\in {\mathbb{R}^{l\times l}}$. I have encountered the following equality: $||v{{v}^{T}}-A||_{F}^{2}={{\left( {{v}^{T}}v \right)}^{2}}-2{{v}^{T}}Av+...
1
vote
3answers
109 views

Equivalency two norms?

Suppose that the following norms on $C^1[0,1]$ . Are they equivalent norms? $\|f\|=\|f\|_\infty+\|f'\|_\infty$ and $\|f\|=\max\{\|f\|_\infty, \|f'\|_\infty\}$ such that $f\in C^1[0,1]$ , $\|f\|_\...
1
vote
1answer
196 views

What is a decreasing scale of Banach spaces?

I am a having a hard time understanding a part of an article I am reading. The screen-cap is below. Basically, it's the line labeled (6) that I do not understand. I am not familiar with the circular ...
1
vote
2answers
1k views

Property of Subordinate Matrix Norm: $\|AB\| \leq \|A\|\|B\|$

I do not understand why the following property for Matrix subordinate norms holds: \begin{equation} \|AB\| \leq \|A\|\|B\| \end{equation} Please explain clearly as I know that it should be shown by ...
1
vote
1answer
195 views

product of bounded linear operators

If I have 2 bounded linear operators $T_1,T_2$ such that $T_2:X\rightarrow Y$ and $T_1:Y\rightarrow Z$. I know that by boundedness, $||T_2(x)||\leq||T_2||\,||x||$ and using the norm of $T$ defined as $...