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

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Operator Norm of a Matrix composed of Standard Basis and Fourier Basis

Let $\mathbf{A}_n$ be an $n\times 2n$ matrix (where $n=2^k$) composed of Fourier basis and standard basis; that is, $$\mathbf{A}_n = \begin{bmatrix}\mathbf{I}_n & \mathbf{F}_n\end{bmatrix}$$ ...
8
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1answer
91 views

For any three vectors $x,y,z\in\mathbb{R}^d$, we have $ \|y-z\|\cdot\|x\|\leq\|x-y\|\cdot\|z\|+\|z-x\|\cdot\|y\|$

Does anyone know a proof of the following problem? Problem: Show that for any three vectors ${\bf x}, {\bf y}, {\bf z}\in \mathbb{R}^d$ the following holds, $$ \|{\bf y} - {\bf z}\|\cdot \|{\bf x}\| ...
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238 views

Finding operator norm

I have to solve the following problem: Find a norm of operator $$A:L^2[-\pi,\pi]\rightarrow L^2[-\pi,\pi]$$ given with $$Af(x)=\int_{-\pi}^{\pi} \cos^2{\left(\frac{x-t}{2}\right)}f(t) \,dt.$$ I ...
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189 views

Sequences are Cauchy depending on the norm?

Assume that we have two Banach spaces $B_1, B_2$ with their underlying sets being identical. Is it possible that a Cauchy sequence in one of the spaces would fail to be Cauchy in the other? I am ...
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256 views

Zero “norm” properties

I have seen the claim that the l0-norm ($\|X\|_0$ = support(X)) is a pseudo-norm because it does not satisfy all properties of a norm. I thought it to be triangle inequality, but am not able to show ...
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708 views

equivalent norms in Banach spaces of infinite dimension

Suppose $ X $ is a Banach space with respect to two different norms, $ \|\cdot\|_1 \mathrm{ e } \|\cdot\|_2 $. Suppose there is a constant $ K > 0 $ such that $$ \forall x \in X, \|x\|_1 \leq ...
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1answer
96 views

Prove that: $\|f\|$ is constant in $G$.

Suppose $G$ is a connected open set of $E$ and $f \in \mathcal{H}(G,F)= \{f: G \to F$, $f$ $\text{is holomorphic mapping} \}$. Suppose there is a points $a \in G$, such that $\|f(x)\| \le \|f(a)\|$, ...
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550 views

Derivation of the polarization identities?

For a real (or complex) inner product space $V$, the inner product can be expressed in terms of the norm as either $$ \langle x,y\rangle=\frac{1}{4}(\|x+y\|^2-\|x-y\|^2) $$ or $$ \langle ...
4
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2answers
367 views

Monotonicity of $\ell_p$ norm

Consider a $n$ dimensional space, it is known (Wikipedia) that for $p>r>0$, we have $$ \|x\|_p\leq\|x\|_r\leq n^{(1/r-1/p)}\|x\|_p. $$ I have two questions about the above inequality. $(\bf ...
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3answers
1k views

Examples of metric spaces which are not normed linear spaces?

Give an example of a metric space which is not a normed linear space. Justify your example.
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65 views

Complex analysis 2: $f \in \mathcal{H}(U,F)$

I have a problem: Suppose $U$ is an open set in $E$ and $f \in \mathcal{H}(U,F)$. Prove that: $1/.$ If $U=E$ then $r_bf(x)=\infty, \forall x \in U$; $2/.$ If $U \ne E$ then $r_bf(x)< \infty, ...
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0answers
37 views

Formulas involving the square of a norm

Why does $$\|x-y+\alpha z\|^2=\|x-y\|^2+2\alpha\langle x-y,z\rangle+\alpha^2 \|z\|^2$$ but $$\|x-z+\theta z-\theta y\|^2=\|x-z\|^2+2\theta\langle x-z,z-y\rangle +\theta^2 \|z-y\|^2?$$ Why is there ...
2
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1answer
392 views

Difference between convergence in norm, point-wise and uniform convergence

I know both definitions but I was wondering what are the relations between them. My question is if someone could explain intuitively the differences between these types of convergence. Specifically, ...
2
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1answer
884 views

Why is the operator norm of a diagonal matrix it's largest value?

I read this in my textbook have tried working through it - I keep getting max 2-norm(Ax), which is just the magnitude of Ax. How should I do this proof? (note, this is not for homework, I'm just ...
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0answers
54 views

Find a matrix that minimizes the norm of a linear function

Let $S=AA^{T}+XX^{T}-AB^{T}\left(BB^{T}\right)^{-1}BA^{T}$ where $A$ and $B$ are known. What is the best possible $X$ that minimizes $\left\Vert S^{-1}y\right\Vert _{2}$ for any vector $y\in ...
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1answer
73 views

Inner product spaces of smooth functions

In the space $C^1([0,1])$ where each $f$ is an element of the space and $$||f||= \left(\int_0^1\left(|f|^2+|f'|^2\right)dx \right)^{1/2}$$ How can it be shown that $||\cdot||$ is a norm of the space?
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598 views

Constrained infinity norm minimization

I have a problem like this: $$\min_x |Ax|_\infty \text{ s.t. } \sum_i x_i = c$$ That is, I want to find the vector $x$ whose elements sum to a constant $c$ that minimized the infinity norm of $Ax$. ...
1
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1answer
61 views

Equality of expressions containing supremum of some double sums

I am doing a project related to operator norm and some double sequences. In the course of proving some results, I encounter the following expressions: $\displaystyle \|\alpha\| := \sup_{\|x\|_p=1} ...
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3answers
366 views

How to prove $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$?

I'm trying to show that $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$. A hint would be nice.
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1answer
59 views

Equivalent continuation of a metric

Hello fellow mathematicians, I am confronted with the following, supposedly not too difficult, problem: Let $(E,f_1)$ be a normed space and $F \subset E$ a linear subspace. Let $f_2$ be a norm on E ...
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190 views

Existence of norm for C*-Algebra

I was wondering wether one can always find/construct a norm which turns an involutive algebra into a C*-algebra. For sure, if it exists it is unique, but does it always exist. If not can you provide a ...
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529 views

Matrix norm induced from vector explanation

Can someone explain as thoroughly as possible what a matrix induced norm is? A concrete example would for this norm would help a lot... $A \in C^{n \times n}, v \in C^n$ $|| A || = ...
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1answer
39 views

Homogenous Systems Question

$A, B \in \mathbb{C}^{n\times n}$ If $I - B$ is singular, then $\exists x$ from $\mathbb{C}^n$ so that $(I - B)x = 0$. How do I show that $|| B || \geq 1$ and if $|| A || \leq 1$ then $I-A$ isn't ...
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Absolute summable sequence in normed space [duplicate]

I have studied that for a sequence of real numbers absolute summability implies summability. What can we say about the sequences $\{x_k\}$ in a normed space . If it is not true in general could ...
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54 views

Is it true that bounded metric can never be induced by norm.

Let $(X, d)$ be a metric space where, $d$ is metric on $X$. We know that metric space $X$ is called bounded if there exists some number $r$, such that $d(x,y) ≤ r$ for all $x$and $y$ in $X$. I want ...
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1answer
67 views

$H^1$ function with smallest seminorm

Let $\Omega\subset\mathbb R^d$ be a Lipschitz domain and $u\in H^1(\Omega)$. Find a $w\in H^1(\Omega)$ with the same boundary values but minimal seminorm on $H^1(\Omega)$. I've read that harmonic ...
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1answer
61 views

Lipschitz condition normed vector space

Am I right that $g: C^{1}[0,2],||*||_{C^1[0,2]} \rightarrow \mathbb{R}$ with $g(f)=f'(1)$ satisfies a Lipschitz condition? Cause $|f'(1)-h'(1)|=|g(f)-g(h)|\le \text{max} |f-h|+ \text{max} |f'-h'|$, ...
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1answer
77 views

Norm of an operator

How to find the norm of the following operator $$ A:\ell_p\to\ell_p:(x_n)\mapsto\left(n^{-1}\sum\limits_{k=1}^n k^{-1/2} x_k\right) $$ Any help is welcome.
4
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1answer
87 views

Square matrix $\|Ax-Ay\|\le \|x-y\|$

Could you give me an example of a square matrix $A\in \mathcal{M}_{2 \times 2}$ or $\mathcal{M}_{3 \times 3}$ for which we have $\|Ax-Ay\|\le \|x-y\|$, $ \ \ x, y \in \{0, e_1, . . . , e_n\}, \ \ e_1, ...
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135 views

Proof of non-strictly convexity of $l_1$ and $l_{\infty}$

Given the Banach space $l_p = \left\{f\in R^n : \|f\| \leq \infty\right\}$ for $1\leq p \leq \infty$, we define the following norms $\|f\|_p = \left(\displaystyle\sum_{j=0}^{\infty} ...
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Proving that Euclidean space having the infinity metric is a complete metric space (stuck)

I am trying to prove that the space ${\mathbb{R}}^k$ with the $\infty$-metric is a complete metric space. I know that I need to show that every Cauchy sequence in the metric space ${\mathbb{R}}^k$ ...
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0answers
85 views

Calculation of the sub gradient of the first norm of a matrix

Lets say I have a matrix X and its first norm $||X||_1$. How do I calculate the subgradient of this norm with respect to matrix X itself.
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1answer
84 views

How to generate a N*D random matrix with columns of unit length?

Is it possible to generate a N*D random matrix with columns of unit length? If not, I also think it is possible of generating a N*D random matrix and, after that, normalizing it in order to have ...
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Understanding weighted inner product and weighted norms

I am reading this book where at page 27 following definitions about weighted inner product and weighted norms are given. Let $M$ and $N$ be Hermitian positive definite matrices of order $m$ and ...
8
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2answers
186 views

Basic question about $\sup_{x\neq 0}{} \frac{\|Ax\|}{\|x\|} = \sup_{\|x\| = 1}{\|Ax\|} $, $x \in\mathbb{R}^n$

I am having trouble with understanding the following definition while studying some basic things related with matrix norms: For every matrix $A\in M_n(\mathbb{R})$ $$\sup_{x\neq 0}{} ...
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0answers
224 views

why is $\ell_0$ a pseudo-norm?

Let $\mathbf{x}$ be a vector in $\mathbb{R}^n$. We define the $\ell_0$ pseudo-norm by:$$\|\mathbf{x}\|_0=\#\left\{i : \mathbf{x}_i\neq0\right\}$$ Why $\|\cdot\|_0$ is not properly a norm?
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Approximation of matrix in 2-norm

The question is the following: Given a matrix $A$ with rank $k$, we are looking for a matrix $B$ of rank $j$, where $j<k$ such that $\|A-B\|_2$ is minimal. My idea was to choose, if $A=P ...
2
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1answer
64 views

Is any norm on $\mathbb R^n$ invariant with respect to componentwise absolute value?

Given $\mathbf{x}=(x_1,...,x_n) \in \mathbb{R}^n$ , define $ \mathbf{x}'=(|x_1|,...,|x_n|) $ . Then, is it $||\mathbf{x}'|| = ||\mathbf{x}||$ for every norm on $ \mathbb{R}^n $ ? NB: The answer ...
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232 views

How to expand equation inside the L2-norm?

I want expand an L2-norm with some matrix operation inside. Assume I have a regression $Y=X\beta+\epsilon$. I want to solve (meaning expand), $$\displaystyle\|Y-X\beta \|_{2}^2$$ Should I do: 1) ...
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1answer
133 views

Inequality for norms

Let g(x, y) be function on $X\times Y$. Show that for all $p\geq q$ $$ \|\,\|g\|_{L^q(Y)}\,\|_{L^p(X)}\leq \|\,\|g\|_{L^p(X)}\,\|_{L^q(Y)} $$ Thsnk you.
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Is it general to say “norm” to mean 2-norm when it is on an inner product space?

Let $V$ be an inner product space over $\mathbb{F}$. If one defines $\lVert \bullet \rVert$ as $\sqrt{\langle \bullet, \bullet \rangle}$, then $\lVert \bullet \rVert$ is a norm on $V$. However, if ...
0
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1answer
287 views

norm of sobolev space $H^{1/2}$

Let $\Omega\subset\mathbb R^d$ a Lipschitz domain and $\Gamma:=\partial\Omega$. For $u\in C^{\infty}(\Gamma)$ we define $$||u||_{H^{1/2}(\Gamma)} = \inf_{\substack{v\in H^1(\Omega) \\ v|_\Gamma =u}} ...
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1answer
54 views

Simple norm equivalence

Please I need help with this problem. Let $V$ be the vector space consisting of functions which are continuous over the interval $[0,1]$, take the value $0$ at the points $0$ and $1$ and are once ...
2
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0answers
59 views

Proving norm inequality with Schwarz's inequality

I'm stuck on the following problem: Suppose that $\{V_j : j \in \mathbb{Z}\}$ is a multiresolution analysis with scaling function $\phi$, and that $\phi$ is continuous and compactly supported. ...
3
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1answer
55 views

Need help to simplify an equation

I am computing an error estimate where at the end I got the following term $\|X_{k} - G\|\leq (q^{2^{k+2}} + q^{2^{k+3}}+ q^{2^{k+4}}....)q^{-3}\|Y_{0}\| + q^{2^{k+1}}.q^{-2}\|X_0\|$ , where $X_k$ ...
3
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176 views

Can the $0$-norm represent determinism?

In Scott Aaronson's Quantum Computing since Democritus, he presents classical probability theory as based on the $1$-norm, and QM as based on the $2$-norm. Call $\{v_1,\ldots,v_N\}$ a unit vector ...
4
votes
3answers
442 views

Norm and square of the ideal $(2,1+\sqrt{-5})$ in the ring of integers

Let $I=(2,1+\sqrt{-5})$ be an ideal of the ring of integers of $\mathbb Q(\sqrt{-5})$. What is its norm $N(I)$? And is $I^2$ principal? My notes say: $1$, $\sqrt{-5}$ is a $\mathbb Z$-basis ...
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Monotone matrix norms

[Ciarlet 2.2-10] Let $\mathscr{S}_n$ be the set of symmetric matrices and $\mathscr{S}_n^+$ the subset of non-negative definite symmetric matrices. A matrix norm $\|\cdot\|$ to be monotone if ...
5
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1answer
207 views

Proving an alternative norm on Sobolev space is equivalent to usual norm

I have this exercice and my problel is only in item 4, and i will desespere. Let $f \in L^2(\mathbb{R}^n).$ 1- Why the equation $\Delta u - u = \dfrac{\partial f}{\partial x_i}$ admits a unique ...
3
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3answers
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Example of two norms on same space, non-equivalent, with one dominating the other

I am looking for an example (with proof) of two norms defined on the same vector space, such that the norms on the two spaces are NOT equivalent, but such that one norm dominates the other...