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

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Maximum of norm

Given a matrix $A$ with $N$ rows and $d$ columns, I would like to prove (or disprove) the following. Let $q(f)=\|(\begin{pmatrix} ...
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What is a bounded sequence of holomorphic functions?

Let $\Omega\subseteq\Bbb C^n$ open, $\{f_n\}_n\subseteq\operatorname{hol}(\Omega,\Bbb C)$ bounded. What does this mean? A numerical sequence $(a_n)_n\subset\Bbb C$ is bounded if $\exists M>0$ ...
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Does this hold for $p=\infty $, i.e., is it true that $(l^{\infty})'= l^1? $ [closed]

Let $E=l^p$ where $1 \le p < \infty $ we know $E'=l^q$ Where $q$ is the dual exponent of $p$, i.e. $q$ is such that $\frac{1}{p}+\frac{1}{q}=1$ Does this hold for $p=\infty $, i.e., is it true ...
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Some problem of function matric space

I have some problem doing my homework Let $|\cdot |$ be define as $|f| = max\{|f(t)|:t\in [0,1]\}$ Define an integral transformation $T:C^0\to C^0$ by$$T(f)(x)=\int_0^xf(t)\,dt$$ (a) ...
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1answer
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What kind of matrix norm satisifies $\text {norm} (A*B)\leq \text {norm} (A)*\text {norm} (B)$ in which A is square?

$||A\times B||\le ||A||\cdot ||B||$ is not always correct. But which kind of matrix norm satisifies this formula for square matrix $A$ and arbitrary matrix $B$?
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69 views

Holder Inequality when $0 < p < 1$ [duplicate]

If $0 < p < 1$, $f \in L^p$, and $\int \lvert g \rvert^q < \infty$, show that $$\int \lvert fg \rvert \ge (\int \lvert f \rvert^p)^{\frac{1}{p}}(\int \lvert g \rvert^q)^{\frac{1}{q}}$$ My ...
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Operator norm and L infinity norm

$g \in L^\infty[0,1]$, $F: L^1[0,1] \to R$ by $F(f) = \int_{[0,1]}fg$, show that $\lVert F\rVert = \lVert g \rVert_\infty$. I can prove that $\lVert F\rVert \le \lVert g \rVert_\infty$ simply by ...
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1answer
46 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 $$\frac{1}{2 \cdot |w|}$$
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Show space C([0,1]) with norm integral is a Banach space [duplicate]

Is the space C([0,1]) with the norm integral from 0 to 1 of |f(t)|dt a Banach space?
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75 views

Expected value of norm of multivariate normal distribution random vector

Let $X$ is a random vector size $p$ from multivariate normal distribution $\mathcal{N}$($0$, $\sigma$ $I$), $I$ is identity matrix. I want to find the expected value of reciprocal of norm like this ...
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“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 $\leq$ ...
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14 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? ...
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55 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 ...
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282 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 ...
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18 views

Explicit example of tensor norms

I can't find any example anywhere on the web where someone actually evaluates a non-trivial tensor norm. So I'm wondering about the simplest non-trivial case. Let $X$ be $\mathbb R^2$ with the ...
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1answer
26 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 ...
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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 ...
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20 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 ...
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32 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 ...
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75 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 ...
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138 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|| ...
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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 ${\bf b}^{\pi} = [b_{\pi(1)}, \dots , b_{\pi(n)}]^T$, ...
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93 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 $\|\cdot\|$ ...
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42 views

Application of the Operator norm $\|.\|_O$ on the differential $df \in \hom(\mathbb{R}^n, \mathbb{R}^m)$

This question origins from my Analysis II Script which gives the following statement (without proof): Lemma Let $U \subset \mathbb{R}^n$ be convex and $f \in C^1(U, \mathbb{R}^k)$ then we have $$ ...
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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 ...
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1answer
32 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? ...
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1answer
51 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 ...
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70 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 ...
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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? ...
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26 views

Pulling p-norm out of sum?

For $\overline x\in \Bbb R^n$ with $||\overline{x}||_p=\left(\sum \limits_{i=1}^n |x_i|^p\right)^{\frac1p}$ Why does $\sum \limits_{i=1}^n \frac{|x_i||y_i|}{||\overline{x}||_p ...
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28 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 = ...
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Dual norm of quasi norms

The dual norm $\Omega^*$ of the norm $\Omega$ is defined for any vector $\mathbf{z} \in \mathrm{R}^N$ by \begin{equation} \Omega^*:= \underset{\mathbf{x} \in \mathrm{R}^N}{max } \quad \mathbf{z}^{T} ...
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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{ ...
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24 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: ...
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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: ...
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Quotient norm question

http://mathoverflow.net/questions/99860/upper-semicontinuity-in-cx-algebras In the 5th paragraph of this post, I don't understand why there exists a vector b satisfying $||a+b||_A < ||q_x ...
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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 ...
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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 ...
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26 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 $w \in W$. ...
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36 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 ...
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the set of Best Approximation is a norm-closed convex set.

Let $V$ be a normed vector space. Let $W$ be a proper closed subspace of $V$. Let $M$ be the set of best approximations in $W$. Prove that $M$ is a norm-closed convex set. I've shown that the ...
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1answer
52 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 ...
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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 $v \in \mathbb R^ 2$ ...
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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|| = ...
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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, ...
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1answer
13 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, ...
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91 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 ...
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37 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 ...