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

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Class of functions is a semimetric space with respect to the $L_2(P)$ metric

I have a doubt relative to the proof of Lemma 19.24 in van der Vaart "Asymptotic Statistics" p.280. Consider the i.i.d. random variables $X_1,...,X_n$. Let $X_i:\Omega \rightarrow \mathcal{X}$ with ...
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41 views

maximum, supremum, and supremum norm

Let $f$ be a non-negative continuous function on a compact metric space $Z$ that satisfies $$\max_{x\in Z } f(x) \cdot \min_{x\in Z } f(x) = 1.$$ Then $$\max_{x\in Z } \log(f(x)) = \left\lVert \log(f) ...
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39 views

For $f \in C(X)$, if $\alpha(f+c)$ belongs to $\overline{\mathcal{A}}$, then $f$ also belongs to $\overline{\mathcal{A}}$

Let $\mathcal{A}$ be an algebra of continuous real-valued functions on a compact space $X$ that contains the constant functions. Let $f \in C(X)$ have the property that for some constant function $c$ ...
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17 views

Approximate decomposition of vector

Let $a,b,c$ be elements of a normed space such that $a+b=c$ and $\lVert a\rVert\leq\lVert c\rVert/2+\varepsilon$ and $\lVert b\rVert\leq\lVert c\rVert/2+\varepsilon$ for some small $\varepsilon>0$. ...
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38 views

Relationship between norm of a function and its supremum

Let $u \in H^{1}(\Omega)$ , $0<\tau <1$ and $B_{\tau R}$ the open ball of radius $\tau R$ under what conditions (I) $||u||_{L^{p}(B_{\tau R})} = \sup\limits_{B_{R}} u$ and (II) ...
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25 views

Equivalence of matrix norms

The equivalence of vector norms on finite dimensional spaces immediately implies that all induced matrix norms are equivalent. However, for matrix norms (like Frobenius norm, Nuclear norm) that are ...
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51 views

Proof related to Hilbert-Schmidt norm

Hi, I am very stuck on this proof and I am not sure how to start it. Any help to get started solving it would be much appreciated.
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38 views

Let $\alpha , \beta \in \mathbb{Z}[\sqrt{d}]$. Prove that $\alpha$ is a unit $\iff N(\alpha) = 1$. [duplicate]

Where $N(\alpha)$ is the norm of alpha. I have proved in a previous question that $N(\alpha\beta) = N(\alpha)N(\beta)$. and I have no trouble proving the $\Rightarrow$ but I don't know how to prove ...
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19 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 ...
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40 views

Matrix induced by p-norm vector defintion

I'm having a bit of trouble understanding the exact definition of a matrix norm that is induced by the vector norm. In this specific case, our matrix norm definition is: $$||A|| = \max\limits_{x \neq ...
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58 views

Does elementwise matrix inequality extend to norms?

The elements of $A$ and $B$ are non-negative and $A_{ij} \leq B_{ij} \; \forall \; i,j$. Is it true that $\Vert A \Vert_p \leq \Vert B \Vert_p$ ? The norm is the operator norm induced by the usual ...
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48 views

What is the norm of this linear operator?

I need to show that $\|A(g_n)\|_1=b-a-\frac1n$ where $A:L^1([a,b])\to L^1([a,b])$ is given by $A(f)(x) = \displaystyle\int_a^xf(t)~dt$, and $g_n:[a,b]\to\mathbb R$ is given by $$g_n(t)= \begin{cases}n ...
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1answer
84 views

What makes a norm “appropriate”? Why can't testfunctions be normed appropriately

I often hear the term "using an appropriate norm". Then I once read that the $C^\infty_0(\Omega)$ cannot be appropriately normed. Why is that? Furthermore, when doing some numerical analysis you often ...
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1answer
53 views

Show that $\langle y_i, y_j\rangle = 0 \forall i \neq j.$

Let $y_1, y_2, . . .$ be a sequence in a Hilbert space. Let $V_n$ be the linear span of $\{y_1, y_2, . . . , y_n\}$. Assume that for $n \ge 1, ∥y_{n+1}∥ \le ∥y − y_{n+1}∥$ for each $y \in V_n$. ...
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42 views

minimum value of ||u-v||, given value of ||u|| and ||v||

given $\|u\|=2$ and $\|v\|=3$ Question: What is the minimum value of $\|u-v\|$? The || sign the the norm How do I go about solving this? $\|u-v\|\le\|u\|+\|v\|$?
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31 views

Squared reverse triangle inequality

Is there a possibility to obtain a bound on the difference of squared norm $\left\lvert\Vert x \Vert^2 - \Vert y \Vert^2\right\rvert$ in terms of the norm of the difference $\Vert x - y \Vert$? I ...
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60 views

Inequality between the norm of derivative and the derivative of norm

Let $x(t)=[x_1(t)~x_2(t)~\cdots ~x_n(t)]^T$, function $x_i:R\rightarrow R$ is differentiable, then it can be drawn that when $p=2$, $\|\frac{d}{dt}x(t)\|_p\geq \frac{d}{dt}\|x(t)\|_p$. I wonder if ...
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21 views

convex envelop on a unit ball

What does that exactly mean by "the function $g$ (continuous) is the convex envelop of the function $f$ (discrete) on a unit $\ell_\infty$ ball". I understand the part of "a function being the convex ...
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42 views

How to prove this inequation (About matrix and vector norm)? [closed]

$\|A\vec{x}\|\leq\|A\|\space\|\vec{x}\|$ where $A$ is a $m\times n$ matrix and $\vec{x}$ is a n-dimensional column vector. Assume that $\|A\|=\sqrt{\Sigma_{i}\Sigma_{j}a_{ij}^{2}}$
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18 views

Norm on following matrices

I know that for infinite, ||A|| = max(absolute values of added first row, second row, third row). But how do you find the for p = 2 and F?
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35 views

Lipschitz continuity of matrix product

Let $x\in\mathbb{R}^n$ and define the matrix functions $A:\mathbb{R}^n\to\mathbb{R}^{m\times p}$ and $B:\mathbb{R}^{n}\to\mathbb{R}^{p\times p}$. Define $F(x) = ...
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36 views

Equivalent norms without Cauchy-Schwarz inequality

Let $X$ be a finite-dimensional vector space over $\mathbb{F}$. ($\mathbb{R}$ or $\mathbb{C}$) Theorem: All norms on $X$ are equivalent. Proof: $a_k$s and $c_k$s will refer to elements of ...
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27 views

$L_p$ norm $\leq L_2$ norm for $1\leq p\leq2$ for Random Variables

Let {$X_i;i\geq0$} be a sequence of random variables defined on the probability space ($\Omega,\mathcal{F},P$). If $||.||_p$ is the $L^p$ norm defined as $||X_i||_p=(E[|X_i|^p])^{1/p}$, how should I ...
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21 views

Can I get the upper bound of the $L_2$ norm of a symmetric matrix?

Suppose there is a symmetric matrix ${\textbf{A}}_{d\times d}$ with each element $a_{ij}$ draw from a Gaussian distribution $\mathcal{N}(0,\delta)$. Is there an upper bound of ${\left\| \textbf{A} ...
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27 views

Converting a norm-computation SemiDefinite program to standard SDP form.

I'm trying to express this norm-computation semidefinite program: for given $A \in R^{m \times n}$ and a scalar $\epsilon \in (0,1)$ $$\gamma_{2}^{\epsilon}(A):= \min\,t\,\, subject\, to\, \left( ...
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38 views

Triangle inequality in norms

Let $N:V\to\mathbb{R}$ be a sequence of norms and $d(x,y)=\sum_{j=1}^{\infty}2^{-j}\frac{N_{j}(y-x)}{1+N_j(x)+N_j(y)}$ Show that $d(x,z)\leq{d(x,y)}+d(y,z)$ I tried the following ...
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35 views

$\sim$ is an equivalence relation on the set of norms on $X$.

Let $X$ be a vetor space. Prove that $\sim$ is an equivalence relation on the set of norms on $X$. Where $\sim $ is the equivalence of two norms. This seems very abstract. What exactly do I have to ...
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65 views

Prove $\|\cdot\|_p$ and $\|\cdot \|_q$ aren't equivalent on $\ell^p$

$1 \leq p < q < \infty$ So we need to find something in $\ell ^p$ that gives different results in each of the two norms but I can't think of anything. I could think of something that is n ...
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1answer
38 views

Prove $\| \cdot \|_{\infty}$ is well-defined on $\ell ^1$

$1 \leq p < q \leq \infty$ $p=1$ and $q= \infty$ Let $(a_k)_{k \geq 1} \in \ell^1$, then $$\sum_{k \geq 1}|a_k|^1< \infty$$ (*) which means $\sup_{k \geq 1 } |a_k|^1 < \infty$. Hence it is ...
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49 views

Prove $\sum \frac{\sin(n^2 t)}{n^2}$ converges

Let $(X,\|\cdot\|)$ be a Banach space and a sequence $(x_n) \subseteq X$ be such that the series $\sum_{n=1}^{\infty} \|x_n\|$ converges in $\mathbb R$. Prove that the sequence $S_n=x_1+\cdots+x_n$ ...
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1answer
51 views

Vector lengths: How to prove this inequality? [duplicate]

I plugged in the summations, tried to square both sides but couldn't reach the desired conclusion. I'm not sure what the second vector would be for the Cauchy inequality. Even when I use the vector ...
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1answer
30 views

Prove $\sup _{t \in [0,1]} |P(t)|$ is a norm

Let $X$ be the vector space containing all polynomials with real coefficients. For every $P \in X$, define $N_1 (P)=\sup _{t \in [0,1]} |P(t)|$ I am having trouble proving the part when you show ...
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Spectral radius of the product of a right stochastic matrix and hermitian matrix

Let us define the following matrix multiplication: $C=AB$ where $B$ is a block diagonal matrix with $N$ blocks, $B_1$, $B_2$ … $B_N$, each of dimensions $M \times M$. I know that $B_k = I_M - \mu ...
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20 views

Spectral radius of the product of a right stochastic matrix and a block diagonal matrix

Let us define the following matrix multiplication: $C=AB$ where $B$ is a block diagonal matrix with $N$ blocks, $B_1$, $B_2$ … $B_N$, each of dimensions $M \times M$. I know that $B_k = I_M - \mu ...
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2answers
103 views

Spectral radius of the product of two matrices

Let us define the following matrix multiplication: $C=AB$ where $B$ is a block diagonal matrix with $N$ blocks, $B_1$, $B_2$ … $B_N$, each of dimensions $M \times M$. I know that $B_k = I_M - \mu ...
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1answer
50 views

Inequality between 1-norm and 2-norm of a vector in [closed]

I have been trying for a while but still can't figure out how to prove this for vectors $x\in\mathbb R^n$: $$||x||_1 \leq \sqrt{n}||x||_2$$ I can prove the case when it's $\sqrt{n}||x||_\inf$ that's ...
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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 ...
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Condition on definition of dual norm

In a review I'm reading, the dual norm is defined as $$||z||_*=\max_{||w||\leq 1}\langle w,z\rangle. $$ Though I'm having a hard time understanding why $||w||\leq 1$ isn't equivalent to $||w||=1$, ...
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Why are matrix norms defined the way they are?

Given $A$ a square matrix Define: $\|A\|_1$ as the max absolute column sum $\|A\|_2$ as the sum of the squares of each element $\|A\|_\infty$ as the max absolute row sum Pray tell, why are ...
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Prove the identity $\| R-A\| < \|Q-A\|$ for all $Q \in SO(3), Q\neq R$?

Let $A$ be a 3 by 3 matrix and positive-definite and let $A=RU$ be its right polar decomposition. How can I prove that $\| R-A\| < \|Q-A\|$ for all $Q \in SO(3), Q\neq R$? Note that $\|.\|$ is ...
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33 views

Find the operator norm

Let $T : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be the linear transformation that the matrix $\begin{pmatrix} 1& a \\ 0 &1\end{pmatrix}$ determines. Find the operator norm $||T||$ with respect ...
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Is it possible to combine L2 norm optimization and thresholding?

Say you have a model like $g = x + \eta$, where $g$ is your observation vector, $x$ the unknown true vector and $\eta$ the additive noise vector. Now you intend to solve for $x$ by $min ||g-x||_2$. No ...
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58 views

Sequence of partial sums converge

Let $(X,\|.\|)$ be a Banach space and a sequence $(x_n) \subseteq X$ be such that the series $\sum_{n=1}^{\infty} \|x_n\|$ converges in $\mathbb R$. Prove that the sequence $S_n=x_1+...+x_n$ converges ...
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40 views

Proving these norms are not equivalent

$X$ is the vector space containting all polynomials with real coefficients. For every $P \in X$, define $N_1(P)= \sup _ {t \in [0,1]} |P(t)|$ and $N(P)=N_1(P)+|P'(1)|$. Prove that $N$ is not ...
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30 views

Norm of a real function

Given a function $g:\mathbb{R}^n\to\mathbb{R}$ which is nonnegative, suppose we take any norm of this function. Is it true to say $$\Vert g(x)\Vert =\vert g(x) \vert = g(x)?$$ Additionally, If we ...
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58 views

Norm in finding local extrema for functional

In "The Calculus of Variations" by Bruce van Brunt, he says: Let $J:C^2[x_0,x_1]\to\mathbb{R}$ be a functional of the form $$J(y)=\int_{x_0}^{x_1}f(x,y,y^\prime)dx,$$ where $f$ is a function ...
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65 views

L2 Norm of Pseudo-Inverse Relation with Minimum Singular Value

Consider a matrix $A \in\mathbb R^{n\times m}$ with $n>m$. It has full column rank, i.e. $\operatorname{rank}(A)=m$. Its left pseudo-inverse is given by; $$A^{-1}_\text{left}=(A^TA)^{-1}A^T $$ ...
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1answer
57 views

$a_n=\int|f|^nd\mu$ then $\lim_{n\to\infty}a_{n+1}/a_n=||f||_\infty$

Let $(X,\mu)$ be a measure space with $\mu(X)=1$. Let $f$ be a measurable function such that $0<||f||_{\infty}< \infty$. Let $a_n=\int|f|^nd\mu$ Show that: (1) ...
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211 views

The 2-norm of the integral vs the integral of the 2-norm

I`m currently having some issues with a seemingly innocent problem. I would like to show that $$\Bigg|\Bigg|\int_\mathbb{R}\begin{pmatrix}A(x)\\B(x)\end{pmatrix}dx\Bigg|\Bigg|_2 \leq ...
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1answer
26 views

Show a bound on the error

Suppose $g \in C[a,b]$, and $M(b-a) < 1$, and let $y$, $y_N$ $\in C[a,b]$ be the unique solutions of the equations, $y = g + Ky$ and $y_N = g + K_N y_N$. $y_N$ has been previously defined as an ...