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 the max can be found only with normalized vectors?

Can someone explain why the term in the first {} equals the second term in the {} :
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Proof of Clarkson's Inequality

Trying to find a proof for Clarkson's inequality, which states that if $2 \leq p < \infty$, then for any $f, g \in L^p$, we have that $$\left|\left|f+g\right|\right|_p^p + ...
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How to represent and handle eigenvalue constraint in optimization

For example consider the problem $\min_X f(X)$ s.t. $\lambda_i(X+A)=\lambda_i(B)$ for $i \in {1,...,N}$ where $A$ and $B$ are full rank N by N matrix, $\lambda_i(X)$ is the i-th eigenvalue of $X$ ...
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is product of norms convex?

Is a function of the form $f(x) = \|x\|_1\|x\|_2$ convex in x? I have tried plotting it in wolfram alpha and it appears convex, althought I ahve not been able to show it yet
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convergence of $L^p $ norm [duplicate]

Possible Duplicate: Limit of $L^p$ norm If I define $|f|_{L^\infty}= \lim_{n\to \infty} |f|_{L^n}$. How can I prove that this limit is esssup $|f|$?
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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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1answer
227 views

Quotient norm on $X\backslash M$

I have $X=(C([0,1]),||.||_1)$ where $||f||_1=\int_{0}^{1}|f(t)|dt$ and $M=\{f\in C([a,b]): f(0)=0\}$. Now I have three questions: 1) Is the quotient norm a norm on the quotient space X\M ? What I ...
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230 views

The Principle of Condensation of Singularities

Let $X$, $Y$ be Banach spaces and $\{T_{jk} : j,k \in\Bbb N\}$ be bounded linear maps from $X$ to $Y$. Suppose that for each $k$ there exists $x\in X$ such that $\sup\{\lVert T_{jk} x\rVert : j ...
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1answer
144 views

Distance between point and linear Space

Suppose $E$ is a normed vector space. Let $f$ be a continuous linear functional on $E$ and denote by $M$ the Kernel of $f$. Let $x\in E$. How to show that ...
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180 views

convergence in $L^2$

Let $ (f_n)$ be a sequence of measurable functions on $ [0,1]$ such that $\int_{0}^1 |f_n|^2 dm \le C$. Assume that there exist a function $f$ such that $\int_{0}^1 |f_n-f|dm \to 0$ as $n \to \infty ...
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why is frobenius norm of a matrix greater than or equal to the 2 norm?

How can you prove that: $$\|A\|_2 \le \|A\|_F$$ I cannot use: $$\|A\|_2^2 = \lambda_{max}(A^TA)$$ It makes sense that the 2-norm would be less than or equal to the frobenius norm but I dont know how ...
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Equivalence of norms is a equivalence relation

Two norms $||-||_1 $, $||-||_2$are equivalent if: for two constants $a,b$ and $x$ from $V$ a vector space over a field it holds that : $$a||x||_1\leqslant ||x||_2\leqslant b||x||_1.$$ This is a ...
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1answer
490 views

Does the limit of a convergent sequence depend on the norm?

Let $X$ be a vector space, and $\|\cdot\|_1$ and $\|\cdot\|_2$ two different (non-equivalent) Norms on $X.$ Let $(x_n)\subset X$ be a sequence and $x\in X$ such that $\lim_{n\to\infty}\|x_n-x\|_1=0.$ ...
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196 views

Proving a matrix equality

I have 2 matrices: $A \in R^{nxn}$ is a non-singular matrix and $B \in R^{nxn}$ is a singular matrix. Here is the expression I need to prove: $$||A - B|| \ge ||A^{-1}||^{-1}$$ I dont understand why ...
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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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1answer
512 views

Inequality of Lebesgue integral with $L^p$-norm

Let $X_t(\omega)$ be a continuous function $t\rightarrow L^p(\omega)$ (i.e., if we fixed the variable $t$ we obtain a function which belongs to $L^p$), with $t\in[0,T]$ and $\omega\in\mathbb{R}$. I ...
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How to prove these three norm equivalence problems

Given $A \in R^{m\times n}$, I have these three norm equivalence equations: $\|A\|_2 \le \|A\|_F \le \sqrt {n}\|A\|_2$ $\frac {1} {\sqrt n}\|A\|_{\infty} \le \|A\|_2 \le \sqrt {m} \|A\|_{\infty}$ ...
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1answer
59 views

Does switching between different $L_p$ norms preserve order?

Suppose you have a list of $n$ dimensional vectors. One can order them by using an $L_p$ norm to do comparisons between vectors. The general questions is, will the order be different depending on the ...
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Two norms on $C_b([0,\infty])$

$C_b([0,\infty])$ is the space of all bounded, continuous functions. Let $||f||_a=(\int_{0}^{\infty}e^{-ax}|f(x)|^2)^{\frac{1}{2}}$ First I want to prove that it is a norm on $C_b([0,\infty])$. The ...
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I am not sure how to calculate this norm?

I have the following matrix: $$A= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{bmatrix} $$ What is the norm of $A$? I ...
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1answer
120 views

How to find if the vector is stable?

Let's say I have the following equation $Ax=b$ My question is - how can i find a vector $b$ around which the above equation is not stable? I have $$A= \begin{bmatrix} 1 & 0.999 \\ ...
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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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81 views

Determine operator norm of mutiplication operator

Consider $$T: (C[-1,1],\|\cdot\|_{2})\rightarrow \mathbb{C}\\Tf :=\int_{-1}^{1}mf\,\mathrm{d}x$$ where $m\in C[-1,1]$. I want to prove $\|T\| = \|m\|_2$. $\|T\|\leq\|m\|_2$ can be easily proved by ...
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1answer
187 views

Equivalence of two norms

Define two norms as following: $$ \left\Vert f\right\Vert _{1}={ \max_{0\leq x\leq1}\left|f\left(x\right)\right|} , \quad\text{ and }\quad \left\Vert f\right\Vert ...
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1k views

How to do the following projection in Matlab?

I have 2 vectors $u$ and $v$ given in $\mathbb R^4$, e.g. $u = (-1,-2,3,4)$ and $v=(1,-2,-3,5)$ I also have $Ax=b$ which is an under-determined system; meaning, if $A$ is $m\times n$, then $m\le n$. ...
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1answer
89 views

How can I give a bound on the $L^2$ norm of this function?

I came across this question in an old qualifying exam, but I am stumped on how to approach it: For $f\in L^p((1,\infty), m)$ ($m$ is the Lebesgue measure), $2<p<4$, let $$(Vf)(x) = ...
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showing that l2 norm is smaller than l1

How can I show that L2<=L1 $||x||_1\le \cdot ||x||_2$ and also $\|x\|_2\leq \sqrt m\|x\|_{\infty}$ regarding the first part, can I say that: $$ \sqrt{\sum\limits_{i=1}^n x^2 } \leq ...
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294 views

Need help with relative and absolute errors?

Lets assume I have $Ax=b$ equation, where $A$ is $2$x$2$ matrix. 1) I want to find an A, x, and b such that relative error in x is small but absolute error in x is large 2) Also want to find A, x, ...
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1answer
98 views

Estimating the integral in norm.

I want to estimate the integral $$\int k(x,y)f(y)dy$$assuming the fact that $k(x,y), f(y)$ are in $L^p, L^q$ respectively. But I want to bound the the whole integral in $L^r$, $r\in [1,\infty]$. I ...
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1answer
219 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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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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1answer
243 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, ...
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109 views

How to calculate norm of operator in Hilbert space

LEt $H_1,H_2$ are two Hilbert spaces. $\{e_1,\ldots,e_n\}\subseteq H_1$ and $\{f_1,\ldots,f_n\}\subseteq H_2$ two orthonormal systems. $\lambda_1,\ldots,\lambda_n\in\mathbb K$. Let $$ U:H_1\to ...
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Confusion about matrix norms

Reading the wiki article I get confused about matrix norms. My question, is it true that $$\lVert Mx \rVert \leq C(\sup_{ij}{M_{ij}}) \lVert x \rVert$$ where $M$ is a matrix and $x$ is a vector and ...
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1answer
220 views

Equivalence between $Lip \ norm$ and $C_1 \ norm$.

Let $f\in C^1([a,b])$. Prove that $\|f\|_{C^1} = \|f\|_{Lip}$. By definition of Lip norm and $C^1$ norm, it is equivalent to prove that $\|f'\|_{\infty}=Lip(f,(a,b))$, where the second member is the ...
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127 views

What does RMSD mean?

Normally a rigid superposition which minimizes the RMSD is performed, and this minimum is returned. Given two sets of points and , the RMSD is defined as follows: $$\begin{align*} ...
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1answer
84 views

Bounding L2 distance with mean and standard deviation

Let $\mathbf{x}=[x_i]_{i=1}^d, \mathbf{y}=[y_i]_{i=1}^d$ be two vectors in $R^d$. Is it possible to find a lower bound $l\leq \|x-y\|$ and an upper bound $u\geq\|x-y\|$ as a function of ...
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1answer
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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Prove unit ball of $L^p(\mu)$ is strictly convex, when $1<p<\infty$

When $1<p<\infty$, $||f||_p=1$,$||g||_p=1$,$f\ne g$,then $\frac{1}{2} ||f+g||_p<1$. I use parallelogram law $||f+g||^2+||f-g||^2=||f||^2+||g||^2=4\\$ Since $f\ne g$, $||f-g||^2>0$ ...
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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 ...
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1answer
479 views

proving $L^\infty$ norm inequality (disprove $\Vert f\Vert_\infty\le\sqrt{n}$)

There are three parts in this question, I've done the first two but not sure about the third one. Also see $L^2$ norm inequality. In the third part, I am asked to show that if $W$ is a ...
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211 views

Find the norm of an operator on $\ell_2$

Let $(x_n) \subset \ell_2$ and let operator $L:\ell_2\to \mathbb R$ be defined by: $\displaystyle L((x_n)) := \sum_{n=1}^{\infty} \frac{x_n}{\sqrt{n(n+1)}}$. Find the norm of L.
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1answer
69 views

Could Ky-Fan Norms Improve the Bound on the Max Norm of $A$

Let $A$ be real symmetric and $D$ shall contain the eigenvalues of $A$. I've learned that $\|A\|_{\text{max}}< \|D\|_{\text{max}}$, where $\|A\|_{\text{max}}$ means the Max norm. I want to get a ...
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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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34 views

How are $\|A\|_{\text{max}}$ and $\|D\|_{\text{max}}$ related?

Max norm The max norm is the elementwise norm with $p = \infty$: $$ \|A\|_{\text{max}} = \max \{|a_{ij}|\}. $$ This norm is not sub-multiplicative. Let $A$ be real symmetric and $D$ ...
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158 views

essential supremum of a matrix multiplication operator

Suppose we have the space $L^p(R,R^n)$ where $1 \leq p < \infty$ (i.e the space of functions that take values in $R^n$ and are $L^p$ integrable) and suppose $T_m: L^p(R,R^n) \to L^p(R,R^n) $ is a ...
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171 views

Norm of operator $g\mapsto \int fg$

Let $T_f:(C([a,b],\mathbb C), \lVert \cdot \rVert_1) \to \mathbb C$ with $g\mapsto \int_a^b f(x)g(x) dx$ for any given $f\in C([a,b],\mathbb C)$. I have to find the norm of $T_f$. I started with: ...
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1answer
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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2answers
348 views

$L^p$ norm of multivariate standard normal random variable

Given $X_i\sim \mathcal{N}(0,1)$ what is the behaviour of $$ ||X||_{l^p}=(\sum_{i=1}^n|X_i|^p )^{1/p}$$ as $n\rightarrow \infty$? For $p=2$ results about $\chi$-distribution tell us that ...
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3answers
160 views

Proving $|x|$ is a norm in $\mathbb {R}^n$

I need some direction on how to start on showing that $| x+y|\leq|x|+|y|$ in $\mathbb R^n$. Note that $$ |x|=\left(\sum\limits_{j=0}^n x_i^2\right)^{1/2} $$ Thank you, Klara