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

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16
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274 views

What does the ideal norm of matrix elements really mean?

Say we have a number field $K$ (specifically, an imaginary quadratic field) and a $2\times2$ matrix $\sigma=\pmatrix{a&c\\b&d}$ with elements $a,b,c,d\in\mathcal O_k$, the ring of integers of $...
8
votes
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524 views

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 $$A\in\...
7
votes
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231 views

renorm a Banach space to make an operator have spectral radius equal to norm

Let $X$ be an infinite-dimensional complex Banach space equipped with the norm $\lVert\cdot\rVert$, and let $T\in\mathcal{L}(X)$ a bounded linear operator on $X$. Let $r(T)$ denote the spectral ...
5
votes
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54 views

Proving linear operator is bounded

Prove that the formula $T(b_1,b_2,b_3,...,b_n,...) = (b_1, b_2/2 ,..., b_n/n ,...)$ defines a bounded linear operator $T : (ℓ^∞,∥·∥_∞)→(ℓ^∞,∥·∥_∞)$. Proving that it is linear is easy. Need help with ...
5
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53 views

Progressed : Convergence problem in Hilbert Space and necessity of inner product

******** PROGRESS : so thanks to Ian's great comment I can get by the proof and that completeness is necessary but I need to know does this hold for general Banach spaces that are not Hilbert spaces? ...
5
votes
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46 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 ...
5
votes
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67 views

Find a relationship between $\hat f$ and $\hat g$, given $g(x)=\int^{x+1}_x f(t)\,dt$, $f\in L^2(\mathbb R)$,

Given $f\in L^2(\mathbb R)$ and $g(x)=\int^{x+1}_x f(t)\,dt$ (a) Relate $\hat f$ and $\hat g$ (b) Prove $g\in H^1(\mathbb R)$ Part A: $$\hat g(k)= \int^{\infty}_{-\infty} \int^{x+1}_{x} f(t)\,...
4
votes
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90 views

A Matrix Norm Inequality $\|A^{1/2}B^{1/2}(A+B)^{-1/2}\|_F \geq \|A^{1/2}(A+B)^{-1/2}B^{1/2}\|_F$

Let $\|X\|_F:= \sqrt{\text{Tr} \left(XX^\dagger\right) }$ denote the Frobenius norm. Does anyone know how to show the norm inequality: $\left\|A^{\frac12}B^{\frac12}(A+B)^{-\frac12}\right\|_F \geq ...
4
votes
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32 views

Equivalences for a Banach lattice

I'm trying to prove the following equivalences for a Banach lattice $E$: $E$ has an order continuous norm Every monotone order bounded sequence in $E$ is convergent E is an ideal in $E^{**}$...
4
votes
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85 views

Differentiation operator on smooth function with compact support

Suppose $f$ is $C^\infty$ with compact support. Let $T_n$ be the operator which sends $f$ to its $n$-th derivative. Is $||T_nf||_\infty$ bounded? It seems like I should use Stone-Weierstrass, but ...
4
votes
0answers
320 views

norm of differential operator on $P^n[0,1]$

Consider the space $P^n[0,1]$ of polynomials of degree $\leq n$ on $[0,1]$, equipped with the sup norm. Now, this is a finite dimensional space, so all linear operators have to be continuous, hence ...
4
votes
0answers
122 views

Proving norm equivalence in $W^{1-1/p,p}(\Omega)$

Define for $p\in [1,\infty)$ and $\Omega=(0,1)^N\subset\mathbb{R}^N$ $$W^{1-1/p,p}(\Omega)=\left\{u\in L^p(\Omega): \ \int_\Omega\int_\Omega\frac{|u(x)-u(y)|^p}{|x-y|^{N-1+p}}dxdy<\infty\right\}$$ ...
3
votes
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36 views

On an inequality concerning vectors with norm less than one

Consider two sets of vectors $\{v_i\}_{i=1}^{n_1}$ and $\{w_j\}_{j=1}^{n_2}$ with $v_i,w_j\in\mathbb{R}^n$ such that $\|v_i\|_2< 1$ and $\|w_j\|_2< 1$ for all $i=1,\dots,n_1$ and $j=1,\dots,n_2$....
3
votes
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39 views

Matrix with roots of unity entries

For a given prime p, i am interested in the norms of matrices which have root of unity entries, i.e., $M_{k,l} \in \{1, \zeta, \dots, \zeta^{p-1}\}$ where $\zeta = \exp{(2\pi I/p)}$. Are there any ...
3
votes
0answers
11 views

Exponentially weighted function in $ \mathcal L_1 $?

I have an interesting adaptive control problem. Consider a signal $u(t)$ generated by normalizing another signal, so that $$ 0 \leq u(t) < 1. $$ Consider the function generated from $u(t)$ as ...
3
votes
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43 views

Proving a trivial bound on $L_2$ norm of the error in a sparse approximation of a vector

Trying to understand this supposedly 'trivial' bound from a paper: If $\theta_N$ denotes the vector $\theta$ with everything except $N$ largest coefficients set to $0$ then we have $$ || \theta - \...
3
votes
0answers
20 views

Finding two adjoints, and showing boundedness of operators

Let $H = l_2$ and consider the following operators: $T,S:H \to H$ $Tx = (0,x_1,x_2,\ldots)$ and $Sx = (x_2,x_3,x_4,\ldots)$ Show they are bounded, and find the adjoint of both: For $T$, I have $\|Tx\|...
3
votes
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23 views

Monotone Convergence Theorem: Application for $p$-norms

Let $X$ be a non-negative random variable in $L^p(\Omega, \mathcal{A},P)$. We have shown that $$||\min(X,M)||_p \leq c$$ for all $M \geq 0$ and want to conclude that $$||X||_p \leq c.$$ In the ...
3
votes
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136 views

Measure theory , Functional calculus, Self Adoint

In $$ L^{2} (\mathbb{R}^2, e^{{-x^2}-y^{2}} dx dy)$$ with subspace $D$ of finite linear combinations of $g_m=(x+iy)^m$ , $m\neq 0$ and integer $(g_0=1)$. I need to show $\langle g_a|g_b \rangle=0$ if ...
3
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85 views

Variational characterization of nuclear norm

The nuclear norm $||\cdot||_{*}$ of a matrix is defined as the sum of its singular values. Working from the result at the bottom of this blog post, we have, for a matrix $\mathbf{X}$ and its ...
3
votes
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38 views

Is the integral form of the polarisation identity useful for anything?

It is well-known that the polarisation identity for real vector spaces is $$ \langle a,b \rangle =\frac{1}{2}\sum_{k=0}^1 (-1)^k\lVert a+(-1)^k b \rVert^2, $$ and the complex generalisation is $$ \...
3
votes
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212 views

Differential Equations: Jordan Form of a Matrix

I am using Lawrence Perko's book Differential Equations and Dynamical Systems, for my Differential Equations course. At the moment we are going over Jordan Forms of a linear system $x^{'}(t) = Ax$, ...
3
votes
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70 views

Weak convergent without completeness implies strong convergence

I want to know if the following holds without completeness: In a normed linear space $H$, $x_n$ is weak convergent to $x$, and $\lim_{n\to\infty} \|x_n\| = \|x\|$ then: $$\lim_{n\to\infty}\|x_n-x\|=...
3
votes
0answers
36 views

Operator norm of $P[v]-P[w]$

Let $\mathcal H$ be a complex Hilbert space with inner product $\langle\mid\rangle$, (dirac notation) which is semi-linear (conjugate linear) in the first argument. Let $\mathcal P_1=\mathcal P_1 (\...
3
votes
0answers
50 views

Operator Norm and Submultiplicativity against the Spectral Norm

Consider $\mathcal{A}:\mathbb{R}^{n\times m}\to \mathbb{R}^{p\times q}$ to be a linear operator. I know that by considering the trace norm and using the submultiplicativity of the operator norm we ...
3
votes
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64 views

Equivalence of norms in $C^1[0,1]$

i have the following problem/questions: I have to prove that $\lVert \cdot \rVert_1 \sim \lVert \cdot \rVert_{*} $ in $C^1[0,1]$; Where $\lVert \cdot \rVert_1$ is the usual $C^1[0,1]$ norm and $\...
3
votes
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40 views

Calculating Norms for the Transpose

How does one show that the transpose mapping $T: \mathcal{M}_{n} \to \mathcal{M}_{n}$, given by $T(a)=a^{t}$, has norm $||T||=1$ but completely bounded norm $||T||_{CB}=n$? Notation. $\mathcal{M}_{n}$...
3
votes
0answers
49 views

Find all $ p \ge 1 $ for which the Hölder norm $\|\cdot\|_p $ is generated by a scalar product.

Find all $ p \ge 1 $for which the Hölder norm $$ \|x\|_p := \left(\sum^{n}_{i=1} |x_i|^p\right)^{\frac{1}{p}} $$ is generated by a scalar product. We know that norm is generated by a scalar product ...
3
votes
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220 views

Rank, nuclear norm and Frobenius norm of a matrix.

The nuclear norm, denoted $\|\cdot\|_*$ is a good surrogate for the $rank$ when minimizing problems like $$\label{pb1}\tag{1} \min_X rank(X) : AX = B $$ Here, we're trying to find an matrix X with low ...
3
votes
0answers
146 views

Do these limits commute?

Given a sequence of functions $f_{n,m}:\mathbb{R}^{n} \to \mathbb{R}$, suppose that $$\displaystyle\lim_{m} f_{n,m}(x)$$ exists almost everywhere (for any fixed n) and also suppose that $$\...
3
votes
0answers
842 views

Operator norm of the inverse

If I made no mistake, one can calculate the operator norm of the inverse of any given (invertible) operator $A: V\rightarrow V$ via: \begin{align}\|A^{-1}\| & = \sup\left\{\frac{\|A^{-1}b\|}{\|b\|...
3
votes
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2k views

Natural matrix norm of an inverse matrix

Let $\left\|\cdot\right\| : \text{GL}(n,\mathbb{R})\to\mathbb{R}_{\ge 0}$ denote the natural matrix norm, i.e. $$\left\|A\right\|:=\max_{x\ne 0}\frac{\left\|Ax\right\|}{\left\|x\right\|}=\max_{\left\|...
3
votes
0answers
52 views

Norms on $\mathbb{Q}$

So with respect to the metric $d(x,y)=|x-y|$ induced by the standard absolute value, the real numbers can be constructed as a completion of $\mathbb{Q}$. With respect to the metric $d_p(x,y)=|x-y|_p$ ...
3
votes
0answers
177 views

Two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ on $X$, and a function $f:X\to Y$ Fréchet differentiable with one of the norms but not with the other one?

There is a theorem that if $f : (X,\|\cdot\|_{X1}) \to (Y,\|\cdot\|_{Y1}) $ is Fréchet differentiable, then replacing the norms with some equivalent norms $\|\cdot\|_{X2}$ and $\|\cdot\|_{Y2}$ ...
3
votes
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204 views

Prove this equality in a Hilbert Space $H$ with Riesz's Representation Theorem.

For $x \in H$, prove that $\sup_{\|z\| = 1} \langle x , z \rangle= \| x \|$ Here is a quick one. If someone could improve this it would be great Proof By Cauchy Schwarz, $\langle x,z \rangle ...
3
votes
0answers
346 views

The norm of an operator

Let $\rho(x)$ be a weight function in a unit sphere, such that \begin{equation} \begin{array}{l} \displaystyle 1. \rho(x)\ge 0,\int_{\mathbb{R}^n}\rho(x)=1\\ \displaystyle 2. \rho(x)\in \mathrm{C}_{0}...
3
votes
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109 views

Exercise from textbook about norm

The below diagram included a relevant page and the question 4 of the exercise which i am not sure how to do. Any hint or solution are welcome Attempts: a) I have done it b) i have tried to show ...
3
votes
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96 views

Is $\rho(f) := \mu(|f| \geq \epsilon), \epsilon >0$ some kind of generalized norm?

Let $\rho(f) := \mu(|f| \geq \epsilon)$ where $\epsilon >0$ is a constant and $\mu$ is a measure and $f$ is a measurable function. Then $f\equiv 0 \rightarrow \rho(f) = 0$ when $|a| \neq 0$, $\...
3
votes
0answers
132 views

Confusion about matrix norms

Reading the wiki article I get confused about matrix norms. My question, is it true that $$\lVert Mx \rVert \leq C(\sup_{ij}{M_{ij}}) \lVert x \rVert$$ where $M$ is a matrix and $x$ is a vector and $C$...
3
votes
0answers
311 views

Minimizing maximum absolute column sum norm of the residual between a matrix and its $k$-rank approximation

Let $X \in \mathbb{R}^{m\times n}$ be a matrix with rank $r$. How can we find the optimal $\tilde{X} \in \mathbb{R}^{m\times n}$ whose rank is $k$ where $k\leq r$ and the reconstruction error in ...
2
votes
0answers
35 views

Polar set of orthogonal matrices set is nuclear norm ball

Reltated problems: Show that the dual norm of spectral norm is Nuclear norm. Proof that nuclear norm is convex. The set of orthogonal matrices is defined as: $$\mathcal{O}(n) = \{X\in \...
2
votes
0answers
43 views

Show that usual $L^2$ norm is equivalent to arbitrary norm $||\cdot||$ that satisfies 'convergence condition'

Given $L^2(\mathbb{R})$ consider a norm $||\cdot||$ on $L^2$ such that $(L^2,||\cdot||)$ is a Banach space and every $||\cdot||$-convergent sequence has a subsequence that converges almost everywhere. ...
2
votes
0answers
20 views

Loosely embedding $\ell_\infty$ in $\ell_2$.

Fix $n$ and $t$. I want a constant $C$ and a function $f : \mathbb{R}^n \to \mathbb{R}^m$ with the following properties. ($m$ can be arbitrary.) If $\|x-y\|_\infty \leq 1$, then $\|f(x)-f(y)\|_2 \...
2
votes
0answers
21 views

Norm of infinite dimensional Hilbert space to calculate difference between string lengths

I am trying to wrap my head around Proposition 13, last para, page 1049 in this paper. The authors are trying to prove certain properties of string edit distance (defined at the start of Section of 6....
2
votes
0answers
22 views

Find value of $p$ such that $\sum |b_n|^p $ converges

Let $(X , \|\cdot \|)=(\ell^2, \|\cdot \|_2)$. Let $e_n \in \ell^2$. Fora bounded linear functional $\phi$ on $X$, let $b_n=\phi (e_n)$ for each $n \geq 1$. Find the value of $p \geq 1$ such that the ...
2
votes
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66 views

Prove that a dual norm is of the same form as the norm in original vector space.

Let $\| \cdot \|$ be any norm on $E^n$. Define on $(E^n)^*$, the dual norm as follows. $$\| a \|^* = \max \{ a \cdot x: \lvert \lvert x \rvert \rvert = 1\}.$$ For every covector $a$, (a) Verify ...
2
votes
0answers
22 views

Norms under Conjugation by Projection Opertaros

I was reading about equivalent forms of the Kadison Singer Problem, and while looking at the Feichtinger Conjecture, I came across the claim that, for a projection operator $P$ and a self-adjoint ...
2
votes
0answers
13 views

Quantifying the angle metric on the Grassmannian in terms of the norm on the exterior power

Let $V$ be a finite-dimensional Hilbert space and $G_k(V)$ the Grassmannian of $k$-dimensional subspaces of $V$. The $k$th exterior power $\bigwedge^k(V)$ can be equipped with a scalar product by ...
2
votes
0answers
14 views

Comparing Euclidean distances using a vector and its projection

Say we have $x\in\mathbb{R}^n$ and $D\in\mathbb{R}$. Define a corresponding vector $y=[y_1\cdots y_n]$ to be the projection of $x$ onto the $n$-cube of side length $2D$ centered at the origin, i.e. we ...
2
votes
0answers
57 views

How did Von Neumann think of the formula for scalar product?

My question is on the idea behind Von Neumann's formula for the scalar product induced from a norm that satisfies the parallelogram law. $\langle x,y\rangle=\frac 14(\|x+y\|^2-\|x-y\|^2)$. Was it by ...