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

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Prove this sequence in $L(\ell_1)$ does not converge to zero?

I have a linear operator $$A: \ell_1 \rightarrow \ell_1$$ $$A_nx=(0,0,...,0,x_{n+1},x_{n+2},...)$$ with $n$ zeros. I am asked to show two things: that for any $x \in \ell_1$, $\lim_{n \rightarrow ...
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127 views

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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46 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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990 views

Frobenius Norm Triangle Inequality

How can I go about proving the triangle inequality holds for the Frobenius norm? I worked through $\|A+B\|_F \le \|A\|_F + \|B\|_F$ and was not able to make it work =/.
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341 views

Frobenius Norm Properties

Say $v\in {\mathbb{R}^{l}}$ (a column vector) and $A\in {\mathbb{R}^{l\times l}}$. I have encountered the following equality: $||v{{v}^{T}}-A||_{F}^{2}={{\left( {{v}^{T}}v ...
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41 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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37 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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111 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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79 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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203 views

Matrix Norm Bounds

A natural consideration of matrix norms is to compare them, and one of the many standard results on the induced 1, 2, and $\infty$-norms indicate that $$\frac{1}{\sqrt{n}}\|A\|_\infty\leq \|A\|_2\leq ...
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388 views

Proving that $\|A\|_{\infty}$ the largest row sum of absolute value of matrix $A$

I am studying matrix norms. I have read that $\|A\|_{\infty}$ is the largest row sum of absolute value and $\|A\|_{1}$ is the highest column sum of absolute values of the matrix $A$. However, I am ...
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326 views

Zero “norm” properties

I have seen the claim that the l0-norm ($\|X\|_0$ = support(X)) is a pseudo-norm because it does not satisfy all properties of a norm. I thought it to be triangle inequality, but am not able to show ...
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66 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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67 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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358 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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244 views

norm equivalence

Exercise 7. of Terence Tao's blog on random matrix, specifically on eigenvalues of Hermitian matrices ...
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89 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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80 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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159 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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194 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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57 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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112 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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84 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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55 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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80 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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1k views

Gradient of squared Frobenius norm

I'd like to find the gradient of $\frac{1}{2} ||X A^T||_F^2$ with respect to $X_{ij}$. Going by the chain rule in the Matrix Cookbook (eqn 126), it's something like $\partial \left[\frac{1}{2} ||X ...
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277 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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171 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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236 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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2k 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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102 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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118 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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697 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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137 views

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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42 views

How to check if a given piecewise defined function on $\mathbb R^2$ is a norm?

I want to check if the function $\parallel (x,y)\parallel = \left\{ \begin{array}{cc} \sqrt{x^2+y^2} & \mbox{if } xy \geq 0 \\ \max\{\vert x\vert, \vert y\vert\} & \mbox{if } xy < 0 ...
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67 views

Show that the sup-norm is not derived from an inner product

I am trying to show that the norm $$\lVert{\cdot} \rVert _{\infty}=\sup_{t \in R}|x(t)|$$ does not come from an inner product (the norm is defined on all bounded and continuous real valued ...
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39 views

Show there exists a Cauchy subsequence

Let $X$ be a separable reflexive real Banach space and $\{\psi_n\}$ be a dense sequence in $$\{\psi\in X' : ||\psi||_{X'} \leq 1\}.$$ Consider in $X$ the scalar product defined by $$(x | y)_0 = ...
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30 views

Gradient of l2 norm squared

Could someone please provide a proof for the following rule: $$\nabla\|x\|_2^2 = 2x$$ I.E. why is the gradient of the $L_2$ norm square of $x$ equal to $2x$? Thanks
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Well-Posedness PDE of the Form $\partial_t u = P(\partial_x) u$ for a Polynomial $P$

My question is to determine whether the PDE $\partial_t u = P(\partial_x) u$, with $2\pi$-periodic boundary conditions, for a polynomial $P$, is well-posed; this depends on the polynomial, and my ...
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25 views

Equivalence of sets

Let $u_1, u_2, u_3 \in \mathbb{C}$ be the cubic roots of unity I'm wondering if the following two sets (balls) are equivalent: $$ \lbrace (v,w) \in \Bbb C^2 : \vert v \vert + \vert w \vert \leq 1 ...
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34 views

Prove or disprove the existence of a length preserving non-normal matrix

Prove or disprove: There exists A such that $\|Av\|=\|v\|$ for any $v \in R^n$, but $A$ is not a normal matrix There exists A such that $\|Av\|=\|v\|$ for any $v \in R^n$, but $A$ is not a unitary ...
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38 views

Show that a set is compact.

Let $X$ be a Banach space and $\{A_t\}_{t\in R}$ a family of linear and continuous maps $X \rightarrow X$ such that function $\mathbb{R} \ni t\rightarrow \|A_t x\| \in \mathbb{R}$ is continuous for ...
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41 views

Find the norm of functional

Consider the functional from $l_2$. $$ x=(x_n)\mapsto \sum \frac{x_n+x_{n+1}}{2^n}. $$ What is the norm of the functional?
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53 views

Show that $\int_{\mathbb R^n}e^{|x|^{-n}}dx=$ Volume of n-sphere

I'm preparing for a calculus exam, I'd like help in solving this question. Let $x \in \mathbb R^n$, $|x|={(x_1^2+x_2^2+...+x_n^2)^{\frac{1}{n}}}$, Show that $$\int_{\mathbb R^n} e^{|x|^{-n}}dx$$ is ...
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38 views

Relationship between matrix 2-norm and orthogonal basis of eigenvectors

Given the following matrix: $$ A = \left( \begin{array}{cc} 3 & 4 \\ 0 & 5 \\ \end{array} \right)$$ calculate $\|A\|_2$, with $\|A\|_2 = max_{x \in \mathbb{R}^2 -\{0\}} \frac{\langle Ax,Ax ...
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46 views

Inequality with a norm

I need help with the following: Let $A=\left(\begin{array}{cc}a & b \\c & d\end{array}\right)$, with $a\in\mathbb{R}$, $b\in(l^{1})^{*}$, $c\in l^{1}$, and $d\in L(l^{1},l^{1})$. Let $h\in ...
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60 views

Why is there an “absolute value” and a norm in the Schwarz Inequality?

This really bothers me, and I'm not sure if it's just that I'm not understanding it correctly. For the moment, assume we are working in a vector space $V$ over $\mathbb{R}^n$. Let $x,y \in V$. We have ...
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109 views

Fredholm operator norm

I have seen here, that the operator norm of a Fredholm operator $T_k(f)(s):=\int_0^1 k(s,t) f(t) dt $, where $k \in L^2([0,1]^2)$ and $f \in L^2([0,1])$ is not equal to the $L^2$ norm of the Kernel. ...
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39 views

Questions over a specific case of the Muntz-Szasz theorem proof

On page 157 of this site: http://arxiv.org/pdf/0710.3570.pdf the author is proving a specific case of one direction of the Muntz-Szasz theorem. I do not understand the following 3 claims: 1) For ...