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

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Why does $\|\sum_{i=1}^{N}\langle f,\phi_i\rangle\phi_i\|^2 = \sum_{i=1}^{N}|\langle f,\phi_i\rangle|^2 $?

I'm reading about Fourier analysis and there is one equality, which I don't understand. Why does: $$\left\|\sum_{i=1}^{N}\langle f,\phi_i\rangle\phi_i\right\|^2 = \sum_{i=1}^{N}|\langle f,\phi_i\...
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1answer
804 views

Frobenius norm is not induced

My textbook says: Frobenius norm defined on $\mathbb{R}^{m,n}$ by a formula: $$\|A\|_F=\sqrt{\sum_{i=1}^n\sum_{j=1}^m|a_{i,j}|^2}$$ when $n>1, \ m>1$ is not induced by any vector's $p$-...
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2answers
37 views

Does $\|A\|_2\leq \|B\|_2$ imply $\|CA\|_2\leq \|CB\|_2$ or $\|AC\|_2\leq \|BC\|_2$?

Assume matrices $A$, $B$, and $C$ are of same dimensions, does $\|A\|_2\leq \|B\|_2$ imply $\|CA\|_2\leq \|CB\|_2$ or $\|AC\|_2\leq \|BC\|_2$? $\|A\|_2$ denotes by $\lambda_{max}\sqrt{A^TA}$, and ...
4
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2answers
158 views

What are some usual norms for matrices?

I am familiar with norms on vectors and functions, but do there exist norms for spaces of matrices i.e. $A$ some $n \times m$ matrix? If so, that would that imply matrices also form some sort of ...
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3k views

Is every normed vector space, an inner product space

Let $V$ be a vector space over $\mathbb{C}$. If $V$ is an inner product space, then $V$ is normed (where the norm is defined as $\|x\|=\sqrt{(x,x)}\,\,$). Now if $V$ is normed, does it follow that $...
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252 views

Is norm non-decreasing in each variable?

Let me try again. Suppose $\|\cdot\|$ is a norm in $\mathbb{R}^n$ and let $$f(x_1,...,x_n)=\|(x_1,...,x_n)\|$$ where $x_i\geq 0, \forall i$. I want to prove or disprove that $f$ is an nondecreasing ...
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1k views

Norm of a Matrix-vector product

Suppose I have vector $\vec x \in \mathbb R^n$ and matrix $\mathbf M$ of dimension $m\times n$. Is there an alternative expression for $\lVert \mathbf M \cdot \vec x \lVert$ that includes $\lVert \vec ...
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1answer
1k views

Norm of the linear functional

Could you help me, please with the following question? There is a linear functional $A : C_{[0;1]} \rightarrow \mathbb{R}$, such that $$ Ax=\int_{a}^{b}x(t)\varphi(t)dt $$ where $\varphi$ is a fixed ...
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1answer
264 views

Equivalents norms in Sobolev Spaces

I know that this is classical but I have never do the calculations to show that the norms in the sobolev space $W^{k,p}(\Omega)$ \begin{equation} \|u\|_{k,p,\Omega}= \Bigl(\int_{\Omega} \sum_{|\alpha|...
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Norm of the operator $T:\ell^2 \to \ell^2$ defined as $(Tx)_1=0, (Tx)_n=-x_n+\alpha x_{n+1}$

Consider the operator $T: \ell^2 \to \ell^2$ defined as $$\begin{cases} (Tx)_1 = 0, \\ (Tx)_n = -x_n + \alpha x_{n+1}, \quad n\ge 2 \end{cases} $$ where $\alpha \in \mathbb{C}$. I want to find ...
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1answer
92 views
4
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62 views

Geometric interpretations of $||z||_p = 1$?

Here $z = a + bi$, with $a, b, \in \mathbb{R}$ and $||z||_p = \sqrt[p]{|a|^p + |b|^p}$. With $p = 1$, this is just diamond (square rotated 45 degrees) of side=$\sqrt2$ centered at the origin. With $...
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135 views

Question about sup norm

Let $x \in \mathbb{R}^n$. Define $|x| = \max\{ |x_1|,...,|x_n|\} $. I want to show that this is a norm on $R^n$. This is my reasoning. First, notice $$ |x| = \max\{ |x_i| \} \geq |x_i| \; \forall i ,\...
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325 views

Poincare Inequality implies Equivalent Norms

I am currently working through the subject of Sobolev Spaces using the book 'Partial Differential Equations' by Lawrence Evans. After the result proving the Poincare Inequality it says the following ...
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163 views

When does $\|x+y\|=\|x\|+\|y\|?$

Let $(V_\mathbb R,\langle,\rangle)$ be an inner product space. I'm trying to see for $x,y\in V$ when does $\|x+y\|=\|x\|+\|y\|?$ Let $\|x+y\|=\|x\|+\|y\|$ Squaring both sides, $\langle x+y,x+y\...
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1answer
59 views

Is the matrix norm of a matrix equal to the maximum of the norms of its Jordan block?

Let $J$ be a Jordan block matrix with blocks $J_1,\cdots,J_n$. I came up with some examples of $J$ and noticed that $\|J\|=\max_{i=1,\cdots,n}\|J_i\|$. Does this result always hold? The norm I use ...
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2answers
143 views

What is the norm of the pre-multiplication by a fixed matrix operator?

Let $A \colon= \left(\alpha_{ij} \right)_{m\times n}$ be a given $m \times n$ matrix of complex numbers, and let the operator $T \colon \mathbb{C}^n \to \mathbb{C}^m$ be defined by $$T(x) \colon= Ax \...
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Proof that the Euclidean norm is indeed a norm

I apologize beforehand for this question. Its embarrassing I know. Anyway, here we go. Recall: $$ \| x \|^2_{\mathbb{R}^2} = \sum^{n}_{i = 1} x^2_{i}$$ How do we prove its a norm? Well if its a ...
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297 views

Inequality between operator norm and Hilbert-Schmidt norm

I have seen the following inequality here but I don't know where I can find a proof for it. Could somebody give me a hint to understand it or guide me to a reference please? $\|AB\|_{HS} \leq \|A\|_{\...
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1answer
173 views

Estimating the sum of a series ($\ell^1$ norm) in terms of two weighted $\ell^2$ norms

Does there exist a $C > 0$ such that $$ \sum_{n \geq 1} a_n \leq C \left( \sum_{n \geq 1} 2^n a_n^2 \right)^{1/4} \left( \sum_{n \geq 1} 2^{-n} a_n^2 \right)^{1/4} $$ for all $a_n \geq 0$ with ...
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93 views

A norm on $\mathbb{R}^2$ such that $\partial C$ is the unit sphere?

Suppose we are on $\mathbb{R}^2$. Assume that $C \subset \mathbb{R}^2$ is a convex bounded neighborhood of the origin invariant by central symmetry. Let $\partial C$ denote the boundary of $C$. My ...
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2answers
121 views

Upper bound for norm of Hilbert space operator

It is a standard result that for a bounded self-adjoint operator $T$ on a complex Hilbert space $H$, we have $||T||=\sup_{||x||=1}|\langle Tx,x\rangle|:=M$. It seems that for any bounded operator on $...
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1answer
57 views

Is it true that $2^p\|f\|^p_{L_p}+2^p\|f'\|^p_{L_p} >2\|f\|^p_{L_\infty}$?

For any $C^1$ function defined in $(0,1)$, is it true that $$ 2^p\|f\|^p_{L_p}+2^p\|f'\|^p_{L_p} >2\|f\|^p_{L_\infty} $$ If it is true, how to prove it?
4
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1answer
156 views

Additive norm $||a+b||=||a||+||b||$

I've read somewhere that there exist spaces where $||a+b||=||a||+||b||$ is true iff $a = \lambda b, \ \ \lambda>0$. Could you tell me what spaces have that property and what spaces don't? $||\...
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1answer
139 views

Counterexample using counting measure

While proving that the norm of the mulplicative operator from $L^2(X) \to L^2(X)$ is the essential supremum of $|g|$ where $g \in L^\infty(X)$, I found that I need the $\sigma$-finiteness of the ...
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5k views

How to prove triangle inequality for $p$-norm?

If $\mathcal{M}=\{M_i : i\in I_n\}$ is a collection of metric spaces, each with metric $d_i$, we can make $M=\prod_{i\in I_n}M_i$ a metric space using the $p$-norm, we simply set $d : M\times M\to \...
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1answer
221 views

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, $$\mathbf{A}_n = \begin{bmatrix}\mathbf{I}_n & \mathbf{F}_n\end{bmatrix}$$ ...
4
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1answer
722 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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Norm of vector in $\mathbb{R}^3$ with multiple

If we have a vector in $\mathbb{R}^3$ (or any Euclidian space I suppose), say $v = (-3,-6,-9)$, then: May I always "factor" out a constant from a vector, as in this example like $(-3,-6,-9) = -3(1,2,...
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1answer
321 views

Finding the norm of the operators

How do I find the norm of the following operator i.e. how to find $\lVert T_z\rVert$ and $\lVert l\rVert$? 1) Let $z\in \ell^\infty$ and $T_z\colon \ell^p\to\ell^p$ with $$(T_zx)(n)=z(n)\cdot x(n).$$...
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1answer
48 views

showing $\|x\| \leq \lim_{n\to \infty} \inf \|x_n\|$

showing $$\|x\| \leq \lim_{n\to \infty} \inf \|x_n\|$$ where $x_n \to x$ weakly, and we are working under a normed space. I am given a hint that $$\|x\| = \sup_{\|\phi\| = 1} |\phi(x)|$$ where $\phi \...
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1answer
71 views

An identity involving matrix norm and vector norm in relation to a density matrix in physics

This question is self-contained. For those who are interested, it arises from my study of the paper: K Lendi (1987), Evolution matrix in a coherence vector formulation for quantum Markovian master ...
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1answer
278 views

Taylor series approximation of function under norm

I am reading this paper. At page number $4$, term $||Au - f||^{2}$ is approximated by taylor series approximation around $u^{k}$. The resulting approximations are $$\|Au - f\|^{2} \approx \|Au^{k}-...
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1answer
114 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
105 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 $(X,\|\cdot\|)...
4
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3answers
164 views

Can one find a stronger norm on a Banach space?

Given a Banach space $V$ of infinite dimension with norm $\|\cdot\|_1$, is that possible to find a norm $\|\cdot\|_2$ on $V$ such that the topology induced by $\|\cdot\|_2$ is strictly stronger than ...
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2answers
124 views

What is the upper bound for this 2-norm

Let $\mathbf{x}$ be the solution to the following problem $$\displaystyle\min_{\mathbf{x}} \|\mathbf{y+Ax}\|_\infty \quad{} \text{subject to} \quad{} \|\mathbf{x}\|_2^2\leq \alpha\|\mathbf{y}\|_2^2$$ ...
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1answer
152 views

Operator norm and Hilbert Schmidt norm

I'm looking for a proof of \begin{equation} ||T||\leq ||T||_{HS}, \end{equation} for which it is sufficient to show \begin{equation} ||Tx|| \leq ||x|| \cdot ||T||_{HS} \forall x\in H, x\not=0 \end{...
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1answer
291 views

Reproducing Kernel Hilbert Space (RKHS) constructed by the summation of positive-definite kernels.

Let $k_1,\ldots,k_p$ be positive definite kernels defined on $\mathcal{X}\times\mathcal{X}$, where $\mathcal{X}$ is a non-empty set. $k_i$ is the reproducing kernel of the Reproducing Kernel Hilbert ...
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2answers
56 views

$\left\{x\in H: 2\leq \|x\|\leq 5\right\}$ is compact?

In a Hilbert space $H$ of dimention infinite, $A=\left\{x\in H:2\leq \|x\|\leq 5\right\}$ is compact? (totally bounded and complete) Thanks in advance.
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1answer
359 views

Metric induced by a norm - what conditions should this metric meet?

$(I)$ I've been browsing some problems concerning metrics not induced by norms, and I've found a comment that said that such a metric should be a concave monotone function. Here is the post I'm ...
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1answer
90 views

Simple question about matrices

My question is simple : If one replaces some of the entries of a matrix by 0, does he obtain necessarily a matrix with a lower norm? I have to precise that the norm I use is the maximum of the ...
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83 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
104 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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2answers
80 views

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 \...
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1answer
382 views

Linear isometry between $c_0$ and $c$

The following question is an exercise and so I'm just looking for advices and not for answers if it's possible. I have the following sets in $l^\infty$ $$c_0 := \{x_n \in l^\infty: \lim x_n = 0\} \...
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1answer
298 views

Proof of an inequality of $L^p$ norms

For a general measure space, we define : $\|f\|_p= \left(\int\vert f\vert^p du\right)^{1/p}$. Let $0 < a < b < c < \infty$ and prove the following: $$ \|f\|_b \leqslant \max\{\|f\|_a, \|f\...
4
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3answers
421 views

Proving two results about the spectral radius

How do I prove these two theorems? Furthermore, can I apply them to infinite-dimensional spaces, such as Banach spaces? Theorem 1. Let $M\in \mathbb{C}_{n\times n}$ be a matrix and $\epsilon >...
4
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1answer
302 views

$\|f+g\|_p\leq\|f\|_p+\|g\|_p$ where $\|f\|_p=(\int_{a}^{b}|f(t)|^p dt)^\frac{1}{p}$

Using the fact that $$cd\leq\frac{c^p}{p}+\frac{d^q}{q}$$ if $$\frac{1}{p}+\frac{1}{q}=1$$ and letting $$c=\frac{|f(t)|}{\left(\int_{a}^{b}|f(t)|^p dt\right)^\frac{1}{p}}$$ and $$d=\frac{|g(t)|}{\left(...
4
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1answer
31 views

Source/explanation for this matrix inequality

Here it is: $$z^\top M^{-1} M^{-1}z \le \|M^{-1}\| z^\top M^{-1} z.$$ Where $\pmb M$ is positive definite symmetric, $z$ is a vector in $\mathbb{R}^p$ (not necessarily normed!) and $||\pmb M||$ is ...