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

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For a normal operator is it true that $\|T^*T^2\| = \|T^3\|$?

For a normal operator is it always true that $\|T^*T^2\| = \|T^3\|$? See the accepted answer for the case in a Hilbert space Update: how about $\|T^*T^2\| = \|T\|^3$ in a Hilbert space
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65 views

Is the max matrix norm induced?

Let $\|A \| = \max_{1 \le i,j \le n} |a_{ij}|$, where $A$ is a square matrix. I can prove that this is a matrix norm, but is it an induced norm?
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195 views

Taking a tricky limit $\lim_{p\to\infty}\int_{\Bbb R^N}\left(\frac{\left\lvert\nabla u\right\rvert}{\left\|\nabla u\right\|_p}\right)^{p-2}\cdots$

$$ \lim_{p\rightarrow \infty} \int_{\mathbb R^N} \left( \frac{\left\lvert \nabla u \right\rvert}{\left\| \nabla u \right\|_p} \right)^{p-2} \frac{\nabla u}{\left\| \nabla u \right\|_p}\cdot \nabla ...
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158 views

Norm of a linear map is not attained

Prove that the norm of the linear functional $$\phi: l^1 \ni \{x_n \} \rightarrow \sum_{n=1} ^{\infty} (1 - \frac{1}{n} )x_n \in \mathbb{K}$$ equals one but there doesn't exist a sequence $ \{x_n \} ...
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1k views

Why is semi-norm special and preferred?

One difference between semi-norm and norm is: "It is possible for $\|v\| = 0$ for nonzero v, $\|\cdot\|$ being semi-norm" I see some papers, and they use semi-norm directly. Why is semi-norm ...
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1answer
2k views

Show that the norm of the multiplication operator $M_f$ on $L^2[0,1]$ is $\|f\|_\infty$

I'm having some (hopefully small) issues computing the norm of an operator. Firstly, the problem, For $f\in L^\infty[0,1]$, define $M_f: L^2[0,1]\to L^2[0,1]$ by $M_f(g)(x) = f(x)g(x)$. Show that ...
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355 views

p-adic norms and products

I came across the following problems about p-adic norms: Problem. Show that $$\prod_{p} |x|_p = \frac{1}{|x|}$$ where the product is taken over all primes $p = 2,3,5, \dots$ and $x \in ...
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1answer
186 views

A name for the property $ \| x \star y \| = \| x \| \| y \| $.

Suppose that $ \star: V^2 \to V $ is some binary operation on a vector space $ V $. Should it hold, is there a name for the following property? $$ \forall x,y \in V: \quad \| x \star y \| = \| x \| \| ...
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63 views

Proving an inequality involving norms of real functions.

If $ f : [a,b] \subset \mathbb{R} \rightarrow \mathbb{R} $ is continuous and differentiable in $(a,b)$, then one can define a norm for such functions as $$ \|f\| = |f(a)| + \max_{x \in (a,b)} |f ...
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Cauchy Sequence some challenge

i read this sentence in one of math books: ‌Every convergent sequence in metric space is a cauchy sequence. would you please some one add more detail, why this is true? thanks.
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237 views

Geometric Mean limit of $\ell_p$ norm of sums

My analysis professor introduced the $\ell_p$ norm to our class as: \begin{align} \| x \|_p = \left(\frac{1}{n}\sum_{j=1}^{n} |x_j|^p\right)^{1/p}. \end{align} We are asked to prove the following: ...
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1answer
1k views

Proof that every normed vector space is a topological vector space

The topology induced by the norm of a normed vector space is such that the space is a topological vector space. Can you tell me if my proof is correct? Of course we have to show that addition and ...
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1answer
40 views

Norms inequality in a sequence space

Let $1 \leq p<q \leq \infty$ (p an q are not related) Let $\Phi$ be the vector space of all sequences with at most finitely many nonzero elements, meaning $\Phi=\{\{x_n\}_{n=1}^\infty|$ there is ...
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1answer
463 views

continuity of norms with respect to $p$

Let $f\in L^{\infty}(\Omega,\Sigma,\mu)\cap L^{1}(\Omega,\Sigma,\mu)$. Then $w(p)=||f||_p$ is continuous function of $p$ for any $p\in [1,\infty)$. How to prove this? I have obtained the proof that ...
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1answer
626 views

A matrix norm inequality

Given a real $m\times n$ matrix $C$, a $m\times m$ diagonal matrix $p$ whose diagonal entries $p_{ii}$ are either 0 or 1, and a $n\times n$ diagonal matrix $q$ whose diagonal entries $q_{ii}$ are ...
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1answer
61 views

Is there any interesting relationship between a Hermitian matrix and its corresponding entrywise absolute?

In general, a Hermitian matrix can have complex off-diagonal terms. Given any Hermitian matrix $[A]_{n,m}$, I can construct another matrix $[\vert A\vert ]_{n,m} =\vert A_{n,m} \vert$. I would like to ...
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1answer
180 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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1answer
97 views

Is there a useful relationship between pointwise and $L^2$ distance?

It would be really convenient to get a bound on the point-wise closeness of functions by knowing their $L^2$ distance. Clearly, if two functions are close in the $L^2$ sense, you cannot get a general ...
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878 views

Equivalence of Frobenius norm and trace norm

According to [1], [2] and other related publications, the following holds for any matrix $X$: $$\| ...
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127 views

Given a matrix $A$ such that $||A||<1$, prove that $I-A$ is invertible

Note: This is a question seen in class while discussing metric spaces and norms, so my recollection might not be 100% accurate. I saw a proof in class, but I wanted to know if there was a different ...
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1answer
206 views

Generalization of $\frac{a + b}{c + d} \leq \text{max}(\frac{a}{c}, \frac{b}{d})$

I'm looking for a matrix version of the basic inequality for the ratio of two sums of positive numbers: $$\frac{a + b}{c + d} \leq \max\left\{\frac{a}{c}, \frac{b}{d}\right\}.$$ Specifically, I have ...
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Dual to the dual norm is the original norm (?)

I have the following questions about dual norms : How do you prove that the dual of the dual norm is in fact the original norm? This is what I have so far: If I have $\|y\|_* $ as the norm dual of ...
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641 views

Norm in a dual space

If $f \in X^*$, with $X^*$ the dual space consisting of all linear bounded functionals on a linear normed space $X$. With the norm defined as $||f||_{X^{*}} = \sup_{||x|| \leqslant 1} |f(x)|$. Why ...
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1answer
470 views

triangle inequality for a certain norm

Let $d$ be a metric on a (say real) vector space $E$, with the property $$d(x,x+cy)=|c|d(x,x+y)$$ for all $x,y\in E$ and scalars $c$. I am trying to prove that $x\mapsto d(x,0)$ defines a norm. The ...
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1answer
899 views

What are norms of sub-matrices invariant under a block diagonal similarity transformation of a block matrix?

Say $M := \begin{pmatrix} A & B\\ C & D \end{pmatrix}$ is a block matrix with $A, D$ being square matrices and this $B$ and $C^T$ having the same shape. Is there any norm characterizing the ...
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1answer
33 views

$X:\Vert X\Vert_2<1 \iff\text{ matrix }\begin{bmatrix} I&X^*\\X&I\\\end{bmatrix} $ is positive

Following question seems so simple, yet I could not come up with a solution. I started to think that there might be sth wrong with the question. Could you please take a look? For a matrix $X:\Vert ...
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1answer
69 views

Does it matter if you use big $L$ or little $l$ when talking about $L$-norms?

I was reading a post on Quora regarding the application of "$l_1$", "$l_2$" norms for convex linear programming when I became very confused at which $L$-norm the posters are actually referring to. I ...
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1answer
59 views

Understanding a part of a proof involving Hilbert-Schmidt norm

I came across a proof I do not seem to understand fully, a screenshot is provided below. my concerns are the following: Why does the fact that $||T||_2 = ||UT||_2$ for every unitary U, allow us ...
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1answer
83 views

A question concerning the Schwartz space

Denote the Schwartz space by $\mathcal S(\mathbb{R})$. I want to show that $\forall n,k \in \mathbb{N} \cup \{0\}$, $\|\cdot\|^{(n,k)} : \mathcal S(\mathbb{R}) \rightarrow [0, \infty)$ defined by ...
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2answers
141 views

For what $X$ do we have $\|XA\|_{1} \leq 1$ for a given $\|A\|_{1} \leq 1$.

All matrices are real. By $\| \cdot \|_1$ I mean a matrix norm induced by the vector norm $L_1$, i.e. the max of the column sums of absolute values. The matrix $A$ is given and we have $\|A\|_{1} ...
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1answer
64 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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1answer
407 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 ...
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1answer
636 views

Using Lax Milgram to find a weak solution in an intersection of Sobolev spaces

I am trying to prove the existence of a weak solution of the problem: $$ -\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0 $$ on the bounded open set $U\subset\mathbb{R}^n$ ...
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834 views

Example of a sequence that converges to two different limits with respect to two complete norms

I've wondered about the following question : Is there an (explicit?) example of a vector space $X$, two complete norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on $X$, and a sequence $(x_n) \subseteq X$ such ...
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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 ...
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1answer
41 views

Corresponding norm from a dual norm?

Let $(X,N_1)$ be a Banach space (separable if necessary) and let $(X^*,N_1^*)$ be its dual space. Here $N_1^*$ denotes the classical dual norm associated to $N_1$. Let $N_2^*$ be an equivalent norm ...
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1answer
175 views

Minimizing the Frobenius norm of a matrix involving the Hadamard product, $\|X(A\odot Y)-S\|_F$

Let $S\in\mathbb{R}^{L\times N}$ and $A\in\mathbb{R}^{M\times N}$ be known and arbitrary. I'd like to solve the following system: \begin{align} \min_{X\in\mathbb{R}^{L\times M},Y\in\mathbb{R}^{M\times ...
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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? ...
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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 ...
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198 views

relation between norms of two vectors

Can we say that; if the $l_1$-norm of an arbitrary vector $a$ is smaller that $l_1$-norm of $b$ ($||a||_1 \le ||b||_1$) then the $l_2$-norm of $a$ is smaller than $l_2$-norm of $b$ ($||a||_2 \le ...
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How do you prove the $p$-norm is not a norm in $\mathbb R^n$ when $0<p<1$?

I see that it fails to satisfy the triangle inequality by example but I don't see how to prove this is the case for all $0 < p < 1$. The definition I am using for $p$-norm is $$ \|A\|_p= ...
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3answers
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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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5answers
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Is closure of convex subset of $X$ is again a convex subset of $X$?

Today I was going through my text book exercise and got hold of the following question. Let $X$ be a normed linear space with norm $\lVert\cdot\rVert$ and $A$ is a non empty convex subset of $X$ ...
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2answers
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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3answers
116 views

Prove that $0 < x < y$ implies $\|x\| < \|y\|$ for any norm.

All vectors are real. Prove that $0 < x < y$ (element-wise) implies $\|x\| < \|y\|$ for any norm. This is probably very basic, but I don't seem to get the hang of it. Edit: it turns out this ...
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4answers
452 views

The difference between $L_1$ and $L_2$ norm?

I have been trying to understand what is the difference between $L_1$ and $L_2$ norm and cant figure it out. In this webpage I got a clear understanding of why we would use $L_1$ norm (scroll down ...
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2answers
154 views

Norm of the integral operator in $L^2(\mathbb{R})$.

Suppose we have an integral operator $A$ such that $$Af(x) = \frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb{R}}e^{-\frac{(x-y)^2}{2}}f(y)dy$$ To find $\|A\|$ we can use the unitary Fourier transform $F$, ...
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3answers
77 views

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 ...
4
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1answer
789 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 ...
4
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2answers
34 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 ...