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

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vector norms involving quantities with a filter relationship

I have two vector norm quantities: $\|\Psi^T(t)\Theta(t)\|$ and $\|\Phi^T(t)\Theta(t)\|$. Here $\Phi^T(t),\Psi^T(t)\in \mathbb{R}^{m\times n}$ and $\Theta(t)\in\mathbb{R}^{n}$. There is a filter ...
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17 views

Existence of a vector whose norm is 1 in a Banach space

Given a finite dimensional Banach space V over reals, I have to show that there exists $v \in V$ such that $\|v\|=1$ At first, I thought that there's an identity element I in V. And $\|I\|=1$. But ...
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When is a norm of identity one?

Is there a specific condition that makes a norm (any norm) of identity equal to one in any Banach spaces? Thanks.
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Given $\mathbf{A}$ stable (all negative eigenvalues), produce a bound on $\|\mathbf{B}\|$ such that…

Given a system: $\dot{\mathbf{x}}=\left(\mathbf{A}+\mathbf{B}\right)\mathbf{x}$ Can you bound $\|\mathbf{B}\|$ s.t. the origin of $\mathbf{x}$ is exponentially stable using a Lyapunov function? ...
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separation in perfectly normal spaces [on hold]

Suppose that we have two distinct points $x$ and $y$ in a perfectly normal compact space. Can we find a norm one continuous function such that $f$ equals to $1$ in some neighborhood of $x$, $f$ equals ...
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1answer
20 views

Trace and Spectral norm of a matrix

Let $A_{n\times n}$ be a matrix. How I can show $$\vert \operatorname{trace} (A) \vert \leqslant n \sqrt{\rho(A^T A)}= n \Vert A \Vert_2$$ and if $A$ is symmetric and positive definite, ...
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1answer
30 views

Operator matrix norm associated with the vector norm

What is the operator matrix norm associated with this vector norm? $$\Vert x \Vert = \frac{1}{n} \mathop {\sum} \limits_{j=1}^{n} \vert x_j \vert, \qquad (x \in \mathbb{R^n})$$
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1answer
52 views

Prob. 14, Sec. 2.7 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $A \colon = [\alpha_{ij}]_{m\times n}$ be a given $m \times n$ matrix of real numbers. Let $\mathbb{R}^n$ be the normed space of all ordered $n$-tuples of real numbers with the norm defined as ...
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How to write Holder's inequality for random vectors?

For $1 < p,q < \infty$ satisfying the constraint $1/p + 1/q =1$ and for $X, Y$ random variables such that $\mathbb E [\vert X \vert ^p ], \mathbb{E} [\vert X \vert ^q ] < \infty $ we have ...
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1answer
43 views

Is it possible that $\Vert I \Vert > 1$ !?

For any matrix norm, is it possible $\Vert I \Vert > 1$ ?, where $I_{n\times n}$ is identity matrix. If not, why in some books they write $\Vert I \Vert \geqslant 1 $ ?
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What is the norm of the operator $\phi: L^3[-2,2] \to \mathbb{C}$ defined by $ \phi(f)=\int_{0}^{1}e^xf(x-1)dx$?

I'm trying to figure out the norm $\|\phi\|$ of the functional $\phi: L^3[-2,2] \to \mathbb{C}$ defined by $$ \phi(f)=\int_{0}^{1}e^xf(x-1)\mathsf dx$$ but am struggling. I can't figure out how to ...
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31 views

$L^2$ vs $L^{\infty}$ norm for interpolation

Under what circumstances should I consider utilizing the $L^2$ norm instead of $L^{\infty}$ when interpolating a function based on sample points? Probably related: Does the answer significantly ...
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1answer
36 views

Why is $A:X\to Y$ linear between two normed spaces is continuous iff bounded?

Why is it that every linear operator $f: \mathbb R^n \to \mathbb R^m$ is bounded and therefore continuous, but why is it that $A:X \to Y$ between two normed spaces is continuous iff bounded? That ...
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1answer
56 views

Find the norm of the operator $T:\ell^2 \to \ell^2$ defined by $Tx := (x_1, x_1+x_2, x_3, x_3+x_4, x_5, x_5+x_6, \ldots)$

Let $T : \ell^2 \to \ell^2$ (involving complex numbers) be defined by $$ Tx := (x_1, x_1+x_2, x_3, x_3+x_4, x_5, x_5+x_6, \ldots). $$ What is $\|T\|$? Essentially I've tried : To find $M \geq 0$ ...
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1answer
80 views

Find the norm of the operator $A:L_2[0,2] \rightarrow L_2[0,2]$ defined by $(Ax)(t) = t \, \operatorname{sgn}(t-1) x(t)$

I have operator: $\boldsymbol{L}_2[0,2] \to \boldsymbol{L}_2[0,2], ( Ax)( t ) = \boldsymbol{t} \operatorname{sgn}(t-1)x(t)$ I need to find operator norm or say that operator isn't bounded. ...
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828 views

Prove that $\|T\|=\sup\{\|T(x)\|: x\in X , \|x\|<1 \}$ for $T$ a bounded linear operator

Let $T: X \to Y$ a bounded linear operator. Prove that $$\|T\|=\sup\{\|T(x)\|: x\in X , \|x\|<1 \}.$$ It is $||T||= \sup \{||T(x)||: x\in X ,||x|| \leq1 \}$ so $||T|| \geq \sup\{||T(x)||: x\in X , ...
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1answer
126 views

How to show that $\|T\|^2=\|T^*T\|$ for a bounded linear operator $T$?

I need to show that for a bounded linear operator, $T$, on a Hilbert space: \begin{align*} \|T\|^2=\|T^*T\| \end{align*} All I have so far: \begin{align*} \|T^*T\|&=\sup\{|\langle T^*Tf,g \rangle ...
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1answer
877 views

Norm of the linear bounded operator $T$ defined by $(Tf)(x) = \int_0^x g(t)f(t)dt$

Some time ago my teacher showed the solution of this exercise. Today I reviewed it, and I think he might be wrong at the last part, c.) Exercise: Let $a > 0$ and let $g \in C[0,a]$ be a ...
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865 views

How can I prove that $\|Ah\| \le \|A\| \|h\|$ for a linear operator $A$?

On http://www.proofwiki.org/wiki/Definition:Norm_(Linear_Transformation) , it is stated that $||Ah|| \leq ||A||||h||$ where $A$ is an operator. Is this a theorem of some sort? If so, how can it be ...
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233 views

What is the norm of the operator $L((x_n)) \equiv \sum_{n=1}^\infty \frac{x_n}{\sqrt{n(n+1)}}$ 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
112 views

How to find the norm of the operator $(Ax)_n = \frac{1}{n} \sum_{k=1}^n \frac{x_k}{\sqrt{k}}$?

How to find the norm of the following operator $$ A:\ell_p\to\ell_p:(x_n)\mapsto\left(n^{-1}\sum\limits_{k=1}^n k^{-1/2} x_k\right) $$ Any help is welcome.
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41 views

Compute $\| T \|$ with the norm $\| \cdot \| := \max_{j=1,\ldots,n} (|\cdot_j|)$

Let $A \colon = [\alpha_{ij}]_{m\times n}$ be a given $m \times n$ matrix of real numbers. Let $\mathbb{R}^n$ be the normed space of all ordered $n$-tuples of real numbers with the norm defined as ...
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1answer
33 views

Upper bound for norm of matrix (cf. Example 2.7-7 in Erwine Kreyszig's book)

Let $A \colon = [\alpha_{ij}]_{m\times n}$ be a given $m \times n$ matrix of real numbers. Let $\mathbb{R}^n$ be the norm space of all ordered $n$-tuples of real numbers with the norm defined as ...
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26 views

is $N(f)=\int_{0}^{1} |f(t)|dt$ a norm on $E$(set of all continous real valued functions defined on [0,1])?

there are three conditions for a norm, I am stuck on one of them which is : $N(f)=0$ iff $f=0$. If i say $\int_{0}^{1} |f(t)|dt=0$ does this imply that $f=0$(zero function) ? Ok the other sense of ...
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63 views

Why does $(\vec{v}\cdot\vec{u})\vec{u}\neq(\vec{u}\cdot\vec{u})\vec{v}$?

I'm trying to save time running Gram-Schmidt. Why doesn't this product equal $||\vec{u}||\vec{v}$? More specifically (and I know this is fundamental and that I should already know it), why doesn't the ...
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32 views

Norm of Fredholm operator in $L^1$

Let $T:L^1([0,1])\rightarrow L^1([0,1])$ be the Fredholm integral operator given by $$ Tf(x)=\int_0^1 k(x,y)f(y)\, dy $$ where $k \in C([0,1]^2)$ is called the kernel of $T$. My problem is to find ...
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1answer
150 views

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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1answer
38 views

Holder's inequality with expectation norm

One defines the ``p-expectation norm" of a function $f$ as $\vert f \vert_p = (\mathbb E ( f^p) )^{ \frac{1}{p}}$. What is the intuition for this norm? Now why are the following true? $\vert ...
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30 views

Frobenius Norm to L2 norm Problem

Here is the problem: if $v^1$, $v^2$, ..., $v^d$ is an orthonormal basis in $\mathbb{R}^d$, then show that $$ ||A - A\sum_{i = 1}^k v^i(v^i)^T ||^2_F = \sum_{i = k+1}^d||Av^i ||_2^2. $$ I am having ...
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25 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 an $m \times n$ 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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1answer
576 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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Compute the value of the Sobolev norm in $H^{-1/2}$

I am working with finite elements using domain decomposition in 2D and one of the solutions I need to obtain is the co-normal derivative of the solution along a segment that is the intersection of two ...
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1answer
25 views

When does the $L_1$ convergence imply almost everywhere convergence?

I know that $L_1$ convergence implies existence of an almost everywhere converging subsequence. But I was wondering, can you tell me some extra conditions on functions that make $L_1$ convergence ...
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1answer
15 views

Infinity Norm calculation $\| \ln(x) - (\ln(3/2) + 2/3(x-3/2)) \|$

I have the following infinity norm: $$ \| \ln(x) - (\ln(3/2) + 2/3(x-3/2)) \|_\infty. $$ Computing from [1,2]. I know that I can compute this in matlab and I get .072. However, how would one go ...
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Differentiation of second norm

Considering $f = {e^{ - {{(||x - y|| - R)}^2}}}$ where ||.|| denotes the second norm, and $x \in {R^n}$. The following derivative is correct? $\frac{{\partial f}}{{\partial x}} = - 2(||x - y|| - ...
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erwin kreyszig introductory functional analysis with applications page 91 exercise 14 [closed]

Let $T$ operator from $X$ to $Y$ and $\dim X=\dim Y= n \le +\infty$ , prove that $R(T)=Y$ if and only if $T^{-1}$ exist I mean number 14
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Norm of orthonormal basis

I know that an orthonormal basis of a vector space, say V is a orthogonal basis in which each entry has unit length. My question is, then, if you have some orthonormal basis say $\{v_1,\ldots,v_8\}$ ...
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1answer
21 views

Supremum vs. Maximum in the definition of the Lp norm [duplicate]

The $L_p$ norm $||A||_p$ is defined as $$\sup_{x \neq 0} \frac{||Ax||_p}{||x||_p} = \max_{||x||_p = 1} ||Ax||_p \tag{1}$$ I'm not quite getting why the LHS uses $\sup$ but the RHS uses $\max$. I ...
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86 views

What are the best books for studying functional analysis in the world

I want to ask you maybe strange question but I really need answer What are the best books for studying functional analysis After Afew week I start study in master so I want references
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28 views

Proving inequality that bounds the sum of norms with the norms of sums (plus additional terms)

I am struggling with showing the following for finite $\delta>0$ and any $g\in\mathcal{G}_1\times...\times\mathcal{G}_k$: ...
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How would I calculate the error in LU Decomposition of a matrix?

I am currently practicing the LU Decomposition of n x n matrices. After finding the LU decomposition, I am lost on how to find the error. I am trying to understand the notation in a problem that asks ...
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36 views

Proof of relation between maximum element and induced $p$-norm of a matrix

If true, prove the identity: $$ ||A|| \ge \max\limits_{i,j}|a_{ij}| $$ $||.||$ is any induced/operator norm. Edit: The identity is true only for operator norm induced by $p$-norm for vectors. I ...
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100 views

Are these two definitions for dual norm equivalent

Suppose there is a $p$-norm $\left\|\cdot\right\|_p$, the dual function at $z$ of it is, $$\sup\left\{ z^{T} x-\left\|x\right\|_p:x\in\mathbb{R}^n\right\}$$ The second is, $$ \sup\left\{ z^{T} x : ...
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Why is this an nuclear norm?

The following figure is from a paper: "The Convex Geometry of Linear Inverse Problems" Just on p.3. As in the figure, the red lines are $2 \times 2$ symmetric unit-Euclidean-norm rank-one ...
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On norms for “more complicated objects”

Whilst studying norms (via various routes) I was always confused by the following definition for some object $\phi$, $$\|\phi\|_p=\sup_{x\in A,x\neq 0}\frac{\|\phi x\|_p}{\|x\|_p}=\sup_{x\in ...
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3answers
66 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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15 views

The meaning of equivalence in norm.

Ask a elementary question: In WIKI: " However all these norms are equivalent in the sense that they all define the same topology." I think "these norms" here mean $l_1, l_2,...l_{inf}$ norms. ...
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1answer
17 views

Norm of a difference of two elements in Quadratic Fields

everyone. Hope you can help me out. I ran across this question and have absolutely no idea how to go about: "Let $X =x+y\sqrt{2}$ with $x,y\in\mathbb{Q}$. Show that there is an element ...
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2answers
37 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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1answer
35 views

Functional analysis, normed space

Can you please help me with this exercise.. I have to check if in interval $[a,b]$ continuously differentiable functions $x=x(t)$ norm can be defined as: $$\vert x(b) - x(a) \vert + \max_{a \leq t ...