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
0answers
40 views

“Transference” argument

In the proof of the Iwaniec-Martin theorem (giving a bound in $L^p$ for the Riesz transform, $\|R_j\|_p=\cot(\frac{\pi}{2p^*})$ the proof of this equality is given by proving the inequalities $\leq$ ...
1
vote
0answers
19 views

condition to be a norm on a finite dimensional complex vector space

Here x is an element of C^n, where n is a natural number and C is the complex number field, and p is any positive number. This becomes a norm when p>=1. But, what happens when p is between 0 and 1? ...
1
vote
2answers
127 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 ...
5
votes
4answers
353 views

Why are we defining the norms on certain vector spaces the way they are?

What's the intuition behind defining $\|x\|_{\infty} = \max_{1 \le i \le n}\{|x_i|\}$ on the space of ordered $n$-tuples of complex numbers? I'm asking because I've been asked to find a norm on the ...
0
votes
1answer
60 views

Is there any shortcuts in getting an H-infinity norm of a matrix expression?

One of the past exam problems I was solving, has this in its official solution: Usually, to calculate the $H_{\infty}$ norm of any matrix expression $M$ I'd first calculate the eigenvalues of ...
0
votes
2answers
30 views

Norm on the space of sequences

Given the sequence spaces $\ell^p$ that are defined as: $$\ell^p = \left\{a = (a_n)_{n\in\mathbb{N}}, \sum_{n=0}^\infty |a_n|^p < \infty\right\}$$ for $\infty > p ≥ 1$, I'm trying to show that ...
0
votes
0answers
26 views

Maximally distant orthogonal matrices

I would like to construct a set of $k$ orthogonal matrices in $\mathbb{R}^{n \times n}$ with maximal summed pairwise distance (in terms of L2 operator norm). Any ideas? I am thinking of just doing ...
0
votes
0answers
61 views

Pseudoinverse with positive solutions

I'm not a mathematician but the engineering problem I'm considering is more of a mathematical question, that's why I post it here: Consider the matrices $M$ ($n \times 1$), $T$ ($n \times m$) and $F ...
3
votes
1answer
102 views

Is the norm operator between normed spaces ever induced from an inner product?

Assume $(V,\| \|_V),(W,\| \|_W)$ are both finite dimensional normed spaces. We have the induced operator norm on $Hom(V,W)$. When does it occur that this norm is actually induced from some inner ...
5
votes
1answer
180 views

Is Frobenius norm induced by 2 vector norms?

Let in the space $V$ defined norm $ ||\cdot||_V $ and in the space $W$ defined norm $ ||\cdot||_W $ Then consider operator norm induced by 2 vector norms $ ||\cdot||_V $ and $ ||\cdot||_W $ $ ||A|| ...
1
vote
0answers
27 views

Order of two vectors to maximise the norm

Given vectors ${\bf a} = [a_1, \dots , a_n]^T$ and ${\bf b} = [b_1, \dots , b_n]^T$, a permutation $\pi$ acting on $[1, \dots ,n]$ and defining ${\bf b}^{\pi} = [b_{\pi(1)}, \dots , b_{\pi(n)}]^T$, ...
1
vote
2answers
180 views

Continuity of metric function

Let $X$ is a vector space and $d$ is a metric function on $X$ and $\|\cdot\|$ is a norm on $X$ and $\langle\cdot,\cdot\rangle$ is an inner product function on $X$ It is to easy to prove $\|\cdot\|$ ...
0
votes
0answers
40 views

The $2$-norm of a Hermitian matrix does not exceed its $1$-norm

How to prove that the $2$-norm of a Hermitian matrix does not exceed its $1$-norm? In wiki, I see $2$-norm of matrix $A$ is $\le \sqrt{\|A\|_1\|A\|_\infty}$. But I don't know how to prove that ...
0
votes
1answer
38 views

Uniqueness of Spatial Median

https://projecteuclid.org/download/pdf_1/euclid.aos/1176350511 Can You help me understand why there is less-than sign in the proof? ...
2
votes
1answer
60 views

Identity of the Operator Norm applied to the differential $df$ for convex $U \subset \mathbb{R}^n$ and $f \in C^1(U, \mathbb{R}^k)$

In my Analysis II Script they often use 'special' norms such as the Operator norm to make proofs more 'elegant' or just shorter. There is also the following statement (without a proof) which I can't ...
4
votes
1answer
101 views

How can I prove that $f$ is inner product function

We know the polarization identity in inner product space : $$\langle x,y\rangle= \frac{1}{4} (\|x+y\|^2-\|x-y\|^2) + \frac{i}{4} (\|x+iy\|^2-\|x-iy\|^2) $$ But the question is if we have ...
1
vote
4answers
320 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? ...
0
votes
1answer
31 views

If a solution to Ax=b does not have any nullspace components, why is it's norm minimum?

I have found a few references on the web stating: If a solution, x, to Ax = b does not contain any nullspace components. Then the euclidean norm is a minimum. My understanding of this is $x + n = ...
0
votes
0answers
70 views

Matrix norm optimization problem : $\min_{\textit{ }x} \| A x B \|_4$, $x$ in the “unit” circle

Bonjour, Let $A$, $B$, $C$ and $D$ complex matrices. Is there a way to find a matrix $x$ (edit: non trivial) as: $\min_{\textit{ }x} \| A x B \|_4^4$ Or, more complicated, $x$ as $\min_{\textit{ ...
0
votes
1answer
34 views

The linearness of extension of linear bounded operator

Let we have the following bounded linear operator $$T: D(T)\rightarrow Y$$ such that $D(T)$ is the domain and it is a vector space and $Y$ is a Banach space . Then it has an extension $$G: ...
-1
votes
2answers
70 views

Some question about extension of bounded linear operator

Let we have the following bounded linear operator $$T: D(T)\rightarrow Y$$ such that $D(T)$ is the domain and it is a vector space and $Y$ is a Banach space . Then it has an extension $$H: ...
-3
votes
1answer
66 views
0
votes
1answer
383 views

How to find the “two norm” of the difference between two vectors

I am using the Jacobi iterative method to estimate the solution to the system of equations $Ax=b$. With an iterative solver you take an initial (educated) guess as to what your $x$ may be, this is ...
1
vote
0answers
44 views

Is the countable product of normed spaces a normed space with this “norm”

Consider the product $\mathcal{N} = \mathcal{N}_{1} \times \mathcal{N}_{2} \times \mathcal{N}_{3} \times \dots$ where $\mathcal{N}_{1}, \mathcal{N}_{2}, \mathcal{N}_{3}, \dots$ are normed spaces with ...
2
votes
1answer
59 views

Prove the set is closed with respect to its norm…

Let $V$ be a normed vector space over R. Let $W$ be a proper closed subspace of $V$. We say $w^*$ is a best approximation in $W$ to $v^* \in V$ if $\|v^*-w^*\| \leq \|v^*-w\|$ for all $w \in W$. ...
0
votes
0answers
49 views

Norm bound for the Jordan basis matrix

Suppose I have a matrix $A$. We can assume, if it will be relevant, that it is stochastic with entries in $\{0,\frac{1}{2},\frac{3}{4},1\}$ and eigenvalues with magnitudes in $(0,1]$. $A$ has a ...
0
votes
1answer
213 views

How to prove a set is norm-closed?

I have to prove that the given space is 'norm-closed convex.' I proved the 'convex' part. But I don't know how to prove a set is 'norm-closed' I think I have to do the followings. Let X be a ...
1
vote
2answers
42 views

Showing existence of a vector $v$ such that $\|Av\|=\|v\|$ given two eigenvalues of the linear transformation $A$

Let $A : \mathbb R ^2 \rightarrow \mathbb R^ 2$ be a linear transformation with eigenvalues $\frac{2}{3}$ and $\frac{9}{5}$. Then, show that there exists a non-zero vector $v \in \mathbb R^ 2$ ...
4
votes
1answer
109 views

Fake proof: Equivalence of norms

Good morning. I'm having a hard time finding what's wrong with the following argument. Let $f$ be any function in $C^{1}([0;1])$ and let $||f||$ and $N(f)$ be two norms defined as follows: $$||f|| = ...
2
votes
1answer
48 views

Norm of Operator Proof

I'm stuck on this problem that I can't seem to figure out. Here's the problem. To note, equation 2.42 says that $$||T|| = \sup \{ ||Tu||: u \in C([a,b]), ||u|| = 1 \}$$ where $T$ is defined, ...
0
votes
1answer
18 views

Relation between a function and its norm

While reading up on Sturm-Liouville system theory, I came across something I didn't fully understand. At one point, in the midst of proving the existence of solutions to the Sturm-Liouvill problem, ...
1
vote
1answer
122 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 ...
0
votes
2answers
45 views

norms of Symmetric Positive Definite Submatrices

Let $\mathbf{A}\in\mathbb{R}^{N \times N}$ be symmetric positive definite. For some $1\leq k<N$, partition $$\mathbf{A}=\begin{pmatrix}\mathbf{A}_{11} & \mathbf{A}_{12} \\ \mathbf{A}_{12}^T ...
0
votes
1answer
70 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 ...
5
votes
0answers
43 views

Matrix equivalence independent of dimension

I'm looking for a criterion on symmetric, positive definite matrices $A$ which ensures the constant $M$ in the lower bound of the norm equivalence $M\|A\|_\infty \le \|A\|_2 \le \|A\|_\infty$ does not ...
0
votes
1answer
44 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 ...
3
votes
1answer
69 views

Is the trace of a matrix a norm?

If the matrix norm of A is defined as $\|A\|=\sum_{i,j}|Aij|$ then how do I determine if the sum of the diagonal elements, i.e., the trace is a valid norm? I am not really sure how to approach this ...
1
vote
1answer
112 views

$H_\infty$ norm of a transfer function by Matlab

Consider the simplest case: $$H(z) = \frac{z+1}{z+2}$$ I use two methods to find $H_\infty$-norm: ...
1
vote
1answer
29 views

Parallelogram law using complex inner product not adding up

Does the parallelogram law still hold in the complex case? Using the following definitions: $\langle \textbf{x}, \textbf{y} + \textbf{z} \rangle = \langle \textbf{x}, \textbf{y} \rangle + \langle ...
1
vote
1answer
33 views

Banach space and its closed subspace. a vectors satisfying inequality.

V=a Banach space over R W=a proper closed subspace of V Prove : For any $\epsilon > 0$, there is a v $\in$_V_ such that ||v||=1 and ||v+w||$\geq$ $1$ - $\epsilon$ I have shown that there exists ...
1
vote
1answer
26 views

vector norms involving quantities with a filter relationship

I have two vector norm quantities: $\|\Psi^T(t)\Theta(t)\|$ and $\|\Phi^T(t)\Theta(t)\|$. Here $\Phi^T(t),\Psi^T(t)\in \mathbb{R}^{m\times n}$ and $\Theta(t)\in\mathbb{R}^{n}$. There is a filter ...
1
vote
1answer
26 views

Existence of a vector whose norm is 1 in a Banach space

Given a finite dimensional Banach space V over reals, I have to show that there exists $v \in V$ such that $\|v\|=1$ At first, I thought that there's an identity element I in V. And $\|I\|=1$. But ...
0
votes
1answer
173 views

Trace and Spectral norm of a matrix

Let $A_{n\times n}$ be a matrix. How I can show $$\vert \operatorname{trace} (A) \vert \leqslant n \sqrt{\rho(A^T A)}= n \Vert A \Vert_2$$ and if $A$ is symmetric and positive definite, ...
0
votes
1answer
104 views

How to write Holder's inequality for random vectors?

For $1 < p,q < \infty$ satisfying the constraint $1/p + 1/q =1$ and for $X, Y$ random variables such that $\mathbb E [\vert X \vert ^p ], \mathbb{E} [\vert X \vert ^q ] < \infty $ we have ...
0
votes
1answer
42 views

Operator matrix norm associated with the vector norm

What is the operator matrix norm associated with this vector norm? $$\Vert x \Vert = \frac{1}{n} \mathop {\sum} \limits_{j=1}^{n} \vert x_j \vert, \qquad (x \in \mathbb{R^n})$$
0
votes
1answer
60 views

Is it possible that $\Vert I \Vert > 1$ !?

For any matrix norm, is it possible $\Vert I \Vert > 1$ ?, where $I_{n\times n}$ is identity matrix. If not, why in some books they write $\Vert I \Vert \geqslant 1 $ ?
1
vote
0answers
54 views

$L^2$ vs $L^{\infty}$ norm for interpolation

Under what circumstances should I consider utilizing the $L^2$ norm instead of $L^{\infty}$ when interpolating a function based on sample points? Probably related: Does the answer significantly ...
1
vote
1answer
53 views

Why is $A:X\to Y$ linear between two normed spaces is continuous iff bounded?

Why is it that every linear operator $f: \mathbb R^n \to \mathbb R^m$ is bounded and therefore continuous, but why is it that $A:X \to Y$ between two normed spaces is continuous iff bounded? That ...
0
votes
1answer
107 views

Prob. 14, Sec. 2.7 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $A \colon = [\alpha_{ij}]_{m\times n}$ be a given $m \times n$ matrix of real numbers. Let $\mathbb{R}^n$ be the normed space of all ordered $n$-tuples of real numbers with the norm defined as ...
4
votes
1answer
61 views

Compute $\| T \|$ with the norm $\| \cdot \| := \max_{j=1,\ldots,n} (|\cdot_j|)$

Let $A \colon = [\alpha_{ij}]_{m\times n}$ be a given $m \times n$ matrix of real numbers. Let $\mathbb{R}^n$ be the normed space of all ordered $n$-tuples of real numbers with the norm defined as ...