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

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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 ...
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1answer
34 views

Some question about linear operator on normed space

$1)$ let $X$ and $Y$ be normed space , show that a linear operator $T:X\rightarrow Y$ is bounded if and only if $T$ maps bounded sets in $X$ into bounded sets in $Y$ $2)$show that the operator ...
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0answers
12 views

Infinity norms and properties

Consider the monic Chebyshev polynomial $$\hat{C}_n(x) = 2^{1-n}\cos{(n\cos^{-1}{x})}.$$ Show, if $Q_n(x)$ is any other monic polynomial of degree $n$, that $$\left\|Q_n\right\|_\infty \ge ...
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1answer
9 views

Infinity norm of the monic chebyshev polynomial

Consider the monic Chebyshev polynomial $$\hat{C}_n(x) = 2^{1-n}\cos{(n\cos^{-1}{x})}.$$ Show that on the interval $[-1, 1]$, $$\left\|\hat{C}_n\right\|_\infty = 2^{1-n}.$$ $(1)$ Now I know we can ...
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1answer
14 views

P Norm in $\mathbb{R}^2$

I am having trouble showing the p-norm is in fact a norm on $\mathbb{R}^2$. We were first asked to show $f(x)=(1-x^p)^{1/p}$ is concave down on $[0,1]$. I now want to prove the triangle inequality ...
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9 views

Are two operator norms on $M_n(A)$ equivalent?

If $A$ is a Banach algebra, then $M_n(A)$ can be given the operator norm as operators on $A\oplus_p\cdots\oplus_p A$ ($1<p<\infty$) to make it a Banach algebra. If in addition $A$ is an operator ...
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1answer
26 views

Matrix norm inducted by infinite vector norm

I'm dealing with a proof form a numerical analysis book whose name is given in the description, I think that I'm missing something very obvious. The problem consists in the second part of the proof, ...
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20 views

Spectral norm linear algebra proof [on hold]

We are using the spectral norm. How does one prove that for any complex matrices $A$, $B$, $C$ with adequate sizes $$ \|A+B\| \le \|A\| + \|B\|, \qquad \|BC\| \le \|B\|\, \|C\|. $$ Thanks for ...
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convergence in different normed spaces

In my class lecture notes, there is such a lemma. Let $X$ be a vector space over $\mathbb F$ and $\lVert \cdot \rVert_1$, $\lVert \cdot \rVert_2$ be two norms on it. If there is $M > 0$ such that ...
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1answer
36 views

Construction of a function $u$ such that $u \in W^{2,2}(\Omega) \cap W_0^{1,2}(\Omega)$ and $u \not\in W_0^{2,2}(\Omega)$

I'm wondering about an example of a function $u \in W^{2,2}(\Omega) \cap W_0^{1,2}(\Omega)$ such that $u \not\in W_0^{2,2}(\Omega)$. Clearly $W_0^{2,2}(\Omega) \subset W^{2,2}(\Omega) \cap ...
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23 views

Bounding cosine of angle between vectors [on hold]

Let $M$ be a symmetric, positive definite matrix such that $0\lt c_1 \le \lambda_{min}(M)\le\lambda_{max}(M)\le c_2$. I am trying to show that $\dfrac{v^TMv}{||Mv||||v||}\gt 0$ for $v\ne 0$ I ...
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0answers
30 views

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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0answers
18 views

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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2answers
47 views

Does this hold for $p=\infty $, i.e., is it true that $(l^{\infty})'= l^1? $ [on hold]

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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1answer
41 views
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1answer
19 views

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
43 views

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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0answers
67 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
42 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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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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21 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 $\leq$ ...
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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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2answers
41 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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4answers
219 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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13 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
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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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0answers
16 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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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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1answer
32 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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1answer
122 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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2answers
60 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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0answers
38 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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24 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 ...
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1answer
27 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
50 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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48 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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23 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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1answer
20 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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15 views

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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1answer
22 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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14 views

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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0
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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 ...