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

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Spectral radius, second induced norm

In my textbook there are few facts left without any sign of a proof, which really bugs me, and I was thinking maybe someone can help me: $A\in \mathbb{R}^{m,n} \Rightarrow \ \|A\|_2 = ...
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55 views

Norm of operator $T_x(f) = f(x)$

Let $X$ be a normed vectorspace and $X'$ be the dual space of $X$. For $x \in X$ we can define $T_x: X' \to \mathbb F$ by $T_x(f) := f(x)$. This is indeed an operator in $X''$. I read that $\| T_x \| ...
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48 views

Norm of a vector is determined by evaluation of linear functionals on it: can this be proved without the Hahn-Banach theorem?

Let $V$ be a normed vector space over the field of real numbers, $\mathbb R$, and let $x_0 \in V$ be fixed. I know how to prove $$\|x_0\| = \sup_{f \in V^*, \|f\| = 1} |f(x_0)|$$ using the Hahn-Banach ...
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128 views

Frobenius Norm with Unitary Operators

For something I'm working on, I have a matrix $A$ with another matrix $U$ which is unitary ($U^*U = I$), and I'm trying to show that, for the Frobenius norm, $\|A\| =\|UA\|$. Now, I can do this ...
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77 views

A normed space of continuous functions with norm $\int_{0}^{1}|f(t)|dt$ is not complete

Suppose $E$ is a normed space of all continuous functions on $[0,1]$ with norm $\int_{0}^{1}|f(t)|dt$. Prove that $E$ is not complete I know that we must do is to find a Cauchy sequence of ...
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85 views

Can $||f|| = ||a||_{q}$ to arbitrary values of $p$ and $q$ satisfying ${1 \over p} + {1 \over q} = 1$

We all know that: Suppose $a = (a_{1}, a_{2}, ..., a_{n})$ is a point in Euclide space $R^{n}$. Consider the mapping $f: R^{n} \rightarrow R$, $f(x) = \sum_{i=1}^{n}a_{i}x_{i}$. Then $||f|| = ...
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263 views

Condition Number of a Product

Is this hypothesis true? $$\operatorname{cond}(AB) \leq \operatorname{cond}(A)\operatorname{cond}(B),$$ where $\operatorname {cond}$ is the condition number. And is this true for rectangular matrices? ...
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4k views

Proof of matrix norm property: submultiplicativity

I've been searching for the definition of the submultiplicative (I think it has multiple names from what I've seen) property in proof form. Some books define it as part of the properties that define ...
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91 views

Prove norm inequality: $\|\mathrm x\|_2 \le \|\mathrm x\|_1$

On $\Bbb R^n$, define for $\mathrm x = (x_1, x_2, \ldots , x_n)$ a norm $$\|\mathrm x\|_1 := |x_1| + |x_2| + \cdots + |x_n|$$ By denoting the usual norm by $\|\mathrm x\|_2$, show that $\|\mathrm ...
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1k views

Norm of a vector-valued function?

When studying commutator estimates, I have encountered the following problem. Consider $f\in C^1(\mathbb{R}^d,\mathbb{R})$ with $\nabla f\in L^p$. So $\nabla f(x)\in\mathbb{R}^d$. My question is ...
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819 views

Convexity of the squared Frobenius norm of a matrix

I was reading this paper where the define an optimization problem as where K and L are kernel matrices and $\pi$ is the permutation matrix. They have explained that the function is convex because ...
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68 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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70 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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763 views

Calculating the Norm of an operator in $L^2(0,1)$

If I have the following operator for $H=L^2(0,1)$: $$Tf(s)=\int_0^1 (5s^2t^2+2)(f(t))dt$$ and I wish to calculate $||T||$, how do I go about doing this: I know that in $L^2(0,1)$ we have that ...
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13 views

Positivity of a map in $(l^\infty(X))^*$

Let $X$ be a set and $\varphi: l^\infty(X)\to\mathbb{R}$ be a linear map such that $||\varphi||=1$ $\varphi(1_X)=1$ I am trying to prove that $\varphi(f)\ge 0$ for all $f\ge 0$, but all my ...
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493 views

norm equivalence

Exercise 7. of Terence Tao's blog on random matrix, specifically on eigenvalues of Hermitian matrices ...
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96 views

Norm of element of Hilbert space

How to prove that in a Hilbert space $H$, $$\lVert h \rVert = \sup_{u \in H}\frac{|(h,u)|}{\lVert u \rVert}?$$ Showing that the RHS is $\leq$ the LHS is easy but not sure of the other part. This is ...
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104 views

On the convexity of the element-wise norm 1 of a pseudoinverse

Let us define $\|A\|_1$ the element wise norm 1 of a matrix $A \in \mathbb{R}^{n \times m}$ as $$ \|A\|_1= \sum_{i,j} |A_{i,j}|. $$ Obviously, this function is convex over $\mathbb{R}^{n \times m}$. ...
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315 views

minimizing a norm and a linear function

Let $y,\lambda\in\mathbb{R}^n$. I want to minimize the following with respect to $y$. $$ f(y)=||y|| + \lambda^Ty $$ where $||y||$ is the Euclidean norm. I first take the derivative of the function and ...
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404 views

Norms of idempotent matrices

So if I have a random idempotent matrix (a matrix that when squared equals itself) how do I go about calculating its 2-norm? I know that a idempotent matrix has eigenvalues of only 0 or 1 and I know ...
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249 views

p norm Matrix relationship

I am trying to show that $\Vert A \Vert_\infty \leq \sqrt{n}\Vert A \Vert_2$ given that $A \in \mathbb{R}^{m \times n}$, $\Vert A \Vert_\infty = \max \limits_{1\leq i \leq n} \sum\limits_{j=1}^n ...
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59 views

Single norm criterion

Let $E$ be a metrizable locally convex space whose topology is defined by an increasing sequence $\{p_n\}$ of seminorms. Show that the topology of $E$ can be defined by a single norm iff there ...
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149 views

norm of a matrix ( which norm have to use ?)

I need to find the norm of the matrix $$ A=\left( \begin{array}{cc} e^{-x} \cos( \sin x) & e^{-x} \sin ( \sin x) \\ -e^{-x} \sin ( \sin x) & e^{-x} \cos (\sin x) \end{array} \right) $$ Here ...
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106 views

Norm properties and completeness

Let $(X,||.||_X)$ be a normed space, M,N two subspaces with norms $||.||_M,||.||_N$ The identity maps are cont. Now I can define the norm $||x||_{M+N}=inf\{||m||_M+||n||_N:m\in M, n\in N, x=m+n\}$ ...
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3k views

prove matrix norm equivalence

Given $A \in R^{m\times n}$, I need to prove: $$||A||_2 \le \sqrt {m}||A||_\infty$$ I have tried a number of things and I just cant seem to get it to work. Also, I need to prove: $$||A||_2 \le ...
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302 views

operator norm and spectral radius

is it true that the operator norm of a matrix $A$ is smaller than 1 if its spectral radius $\rho(A)$ is smaller than 1? many thanks for any help, it is much appreciated!
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75 views

Norm of two operators

(1) $U(x)=a \langle x,b \rangle +b \langle x,a \rangle $, $a,b\in H\setminus \{0\}$. U is an operator from H to H and a,b are orthogonal elements. I want to calculate $||U||$ For this one I tried the ...
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87 views

How to show that the scalar product on a vector space extends by continuity to a scalar product on the completion of the vector space?

I'm trying to solve the following problem: Assume $H_0$ is a vector space equipped with a scalar product. Complete $H_0$ with respect to the norm $\Vert x \Vert = \langle x,x \rangle^{1/2}$. We ...
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155 views

what does it mean to say a space is norm separable?

I came across in my textbook the term: norm separable. I looked in the textbook and online and could not find a definition.
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454 views

Do Lipschitz-continuous funcions have weak derivatives on bounded open sets?

Let $\Omega\in\mathbb{R}^n$ be open and bounded. I'm wondering if a function $f\in C^{0,1}(\Omega)$ (a Lipschitz-continuous one) is also an element of $W^{1,2}(\Omega)$ (that is the space of weakly ...
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224 views

Different norms for space $C[0,1]$

For the space of all continuous functions we can have the sup norm: $|f|=\sup|f|$ I have also seen the following norm: $|f|=\sup|f(x)|/|x|$ I don't know what this norm is called and therefore can't ...
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259 views

Looking for an inequality related to the Cauchy-Schwarz inequality

From the Cauchy-Schwarz inequality, we can prove that $$\lVert w(x)\rVert^2_{L^2_{[0,1]}}=\int_0^1 w(x)^2\, dx \leq \sqrt{\int_0^1 w(x) \,dx}\cdot \sqrt{\int_0^1 w(x)^3\, dx}.$$ Is it possible to ...
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3k views

$L_{2}$ norm of the gradient of a vector valued function.

I have a vector valued function $U(x,y)=\Big(u_{1}(x,y),u_{2}(x,y)\Big)$. I want to find $\|\nabla U\|_{L_{2}(0,1)}$, but i could not figure how can do it. Do you have any idea?
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108 views

Equivalent norms on $\mathbb{R}^2$

For $(\mathbb{R}^2,\|\cdot\|_2)$ and $(\mathbb{R}^2,\|\cdot\|_\infty)$ and any $x \in B((0,0),1,\|\cdot\|_2)$ how would you find a $\delta_x$ such that $B(x,\delta_x,\|\cdot\|_\infty) \subset ...
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157 views

Question about norms of a matrix when exchanging two of its rows

Assume I exchange two rows of a square complex $n\times n $ matrix. Are the Euclidean norm and the Hilbert-Schmidt norm of the new matrix (obtained from the first one by exchanging two of its rows) ...
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2k views

Cauchy sequence in a normed space

Let $V$ be a real vector space. Suppose that $\Vert \cdot \Vert_1$ and $\Vert \cdot \Vert_2$ are two norms on $V$ which are equivalent. I suspect the following to be true. Let $(x_n)_{n=0}^\infty$ ...
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Norms for space $V$ of vector-valued mappings $f: \mathbb R^m \rightarrow \mathbb R^n$

I wonder whether there are natural norms for the space $V$ of vector-valued functions that map $\mathbb R^m$ into $\mathbb R^n$. Formally, let's define $V$ as the set of $f$ such that $f: \mathbb R^m ...
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30 views

Is the result true when the valuation is trivial?

Here I proved the following result: Proposition: Let $(K,|\cdot|)$ be a valued field with non-trivial valuation and let $X$ be a vector space over $(K,|\cdot|)$. Two norms $p_1,p_2$ on $X$ are ...
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35 views

Prove that $(V,\lVert\cdot\rVert_\infty)$ is a Banach space

$X=C[0,1]$ (i) Prove that if the sequence $(f_n)_{n\ge1} \subseteq X$ converges to $f \in X$ in the supremum norm, then for each $t\in[0,1]$ one necessarily has $\lim_{n\to\infty} f_n(t) = f(t)$. ...
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Norm of rows of a matrix with a given spectral norm

Given a $n \times n$ matrix $M$ such that $\|M\|_2 = c$ (where $\|\|_2$ denotes spectral norm, or operator norm), is it true that for all $i = 1...n$ it holds that $$\sqrt{(\sum_{k=0}^n M(i,k)^2)} ...
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29 views

Spectral Norm Proof

I don't really understand the question below: A matrix can have small eigenvalues but large spectral norm (i.e., largest singular value). Find such a matrix A of proper dimension so that all of its ...
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32 views

Find $\|T\|_{\ell^\infty \rightarrow \ell^2}$ [duplicate]

Let $(v_n)_{n \geq 1} \in \ell ^2$ be a fixed bounded sequence of real numbers. Define a mapping $T$ on $\ell^\infty$ using the formula $$T(a_1, a_2,\ldots)=(v_1a_1,v_2a_2,\ldots), \qquad ...
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18 views

Show $||Tx||_2 \leq ||(v_n)||_2 \cdot ||x||_{\infty}$

Let $(v_n)_{n \geq 1} \in \ell ^2$ be a fixed bounded sequence of real numbers. Define a mapping $T$ on $\ell ^{\infty}$ using the formula $$T(a_1, a_2,...)=(v_1a_1,v_2a_2,...), \, \, \, \, \, ...
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21 views

Inner Product on Sobolev Space with p=2

Wikipedia defines the Sobolev Space: $H^{s,p}(\mathbb{R}^n)= \left\{f \in L^p(\mathbb{R}^n): \mathcal{F}^{-1}[(1+|k|^2)^{\frac{s}{2}} \mathcal{F}f] \in L^p(\mathbb{R}^n) \right\}$ Where $s \in ...
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25 views

Closedness and continuity in infinite dimensional spaces

I cannot understand why the operator $A=d/dx: D(A)(\subset C[a,b])\to C[a,b]$ is closed when the domain $D(A)$ is chosen to be $C^1[a,b]$ while we know that we can converge to a non-differentiable ...
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53 views

Proof of a p-norm and Cauchy/Cauchy-Schwarz inequality generalization

I've started to study metric spaces recently and got intrigued with a demonstration which led me to a bunch of ideas and questions. First of all, I've proved the Cauchy inequality and then ...
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24 views

Does concept of submultiplicative norm make sense for non-square matrix?

I am confused by conflicting answers to the question. Some books seem to define submultiplicativity of matrix norms only for square matrices while some books dont mention such a restriction. On this ...
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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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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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52 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 $y \in V_n$. ...