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

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42 views

Prove Operator Norm is a Norm on linear space [duplicate]

Prove that the operator norm defined by $$\left \| A \right \| = \left \| A \right \|_{V\rightarrow W} = \sup_{0\neq v\in V} \frac{\left \| Av \right \|_{W}}{\left \| v \right \|_{V}}$$ (Given norms ...
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26 views

Convergence of sequence of discrete measures on $\mathbb{T}$ in total variation norm

Let $\mathbb{T}=\mathbb{R/Z}$ be the circle. Prove that the space of discrete measures on $\mathbb{T}$ is closed under convergence in total-variation norm in the set of measures on $\mathbb{T}$. ...
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26 views

Equivalence of norms on $W^{1, p}(I)$

Let $I=(a, b)$, $a<y<b$ an arbitrary point and $J\subseteq I$ an open interval. Then the norms $\left\Vert \cdot \right\Vert_{\sharp}$, $\left\Vert \cdot \right\Vert_{\flat}$ on the Sobolev ...
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0answers
136 views

Measure theory , Functional calculus, Self Adoint

In $$ L^{2} (\mathbb{R}^2, e^{{-x^2}-y^{2}} dx dy)$$ with subspace $D$ of finite linear combinations of $g_m=(x+iy)^m$ , $m\neq 0$ and integer $(g_0=1)$. I need to show $\langle g_a|g_b \rangle=0$ if ...
1
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1answer
18 views

Defining a distance between images

Let's consider raster images represented by bi dimensional real matrices. I have an original image $M_0$, and after transforming it several times I get a set of related images $M_n$, which have the ...
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0answers
24 views

Compute of two norms of a function of three variables

Let $f$ be a function defined on $\mathbb R^3$ by $$f(x,y,z)=\exp(-2\mathbb i\pi (x+y+z)) |x|^{1-k} |y|^{k-1} \operatorname{sign}(x) \operatorname{sign}(z),$$ where $sign(x)$ means the sign of $x$ and ...
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1answer
39 views

Find vectors x and y with given norms.

I've spent many hours on this and I just can't understand how to do this. Could you please go through this with me? I have a test, and I really need to understand how to do these types of problems. ...
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2answers
28 views

What is the norm of matrices? Is it related to the norms of linear transformations? [closed]

What are the norms of a matrix? Is there any relation with norm of linear operators/transformations?
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1answer
27 views

Equivalency of two norms

Let $U$ be a normed vector space with two norms: $ || . ||,|| . ||^{'} $ For every sequence $\{x_n \}$ that $||x_n-x||\rightarrow 0 $ & $||x_n-y||^{'}\rightarrow 0 $,we can conclude $x=y$. ...
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1answer
39 views

the relation between cardinality, L1-norm and L2-norm of a vector

For every $u\in \mathbb{R}^n$, $\textbf{Card}(u)=q$ implies ${\lVert u \rVert}_1 \leq \sqrt{q} {\lVert u \rVert}_2$ where $\textbf{Card}(u)$ is the number of non-zero element (so the L0-norm). Why ...
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1answer
38 views

Showing that $\{\sin(nx)\}_{n=1}^\infty$ is a complete orthogonal system in $C([0,2\pi])$ and $L_2[0,\pi]$.

Showing that $\{\sin(nx)\}_{n=1}^\infty$ is a complete orthogonal system in $C([0,2\pi])$ and $L_2[0,\pi]$. So set the inner product for $C([0,2\pi])$ to be $\langle u,v\rangle = \int_0^{2\pi} ...
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0answers
9 views

Lattice points with next-largest norm

In a 2D integer grid, the points in increasing distance from the origin are: $(0,0)$ $(\pm1,0)$ and $(0,\pm1)$ $(\pm1,\pm1)$ etc By symmetry we need only consider one-eighth of the lattice, $x\ge0$ ...
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1answer
40 views

Show a continuous function on a closed bounded interval is Lipschitz under the maximum (infinity) norm

I'm currently working on the following: Define the function $\psi: C[a,b] \to \mathbb{R}$ by $\begin{equation*} \psi(f)=\int_{a}^{b}f(x)\,dx \end{equation*}$ for each $f \in C[a,b]$. Show that ...
1
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1answer
24 views

Show which of the following norms are equivalent

On the vector space $C^1[0,1]$ of all real valued continuously differentiable functions defined in $[0,1]$, consider the following norms : $\displaystyle ||f||_{\infty}=\sup_{0\le x\le 1}f(x)$ , ...
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30 views

Why does the equality hold?

For $A \in \mathbb{R}^{n \times n}$ with $A=A^{T}$ we set: $$\lambda:= \max \{ \langle x, A x \rangle: ||x||_2=1\} \\ \mu:= \min \{ \langle x,A x \rangle: ||x||_2=1\}$$ Then for $x \in ...
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1answer
40 views

Is $|(v,\frac{Pv}{||Pv||})|=||Pv||$ when $P$ is an orthogonal projection?

Suppose $P$ is an $k \times k$ matrix that represents an orthogonal projection. Let $v$ be an $k \times 1$ vector. Let the operator $(\cdot,\cdot)$ represents the scalar product. Does this ...
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0answers
29 views

Correct this solution to finding $||A||$ of $(Af)(x) = g(x)f(x)$.

I need some help with a question I tried to solve it, but I am just not quite sure if my answer is correct. (I have got the feeling it can be - much - better). Suppose we have a complex Hilbert space ...
0
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1answer
38 views

Matrix norm inequality proof: inverse of two p.s.d matrices sum

I wonder if the following matrix norm inequality holds: Let $A$ and $B$ are both strictly symmetric positive definite matrix $\|(A+B)^{-1}\|_2\leq \|A^{-1}\|_2$ ? Thanks in advance.
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23 views

Relation of the upper triangular factor and the original matrix

Suppose $$PA = LU$$ is the LU factorization(exact) of the square real matrix A, L is the unit lower triangular matrix. Is there a way to determine the relation between the norm of $U$ and the norm ...
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1answer
21 views

Norm of $T^n$, where $Tf(x,y) = \begin{cases}f(x+y/b,y), &0<x<1-y/b,\\1/2f(x+y/b-1,y),& 1-y/b<x<1.\end{cases}$

Let $0 < a < b$ and $T\colon L^\infty((0,1)\times (a,b)) \to L^\infty((0,1)\times (a,b))$ be the operator defined by $$Tf(x,y) = \begin{cases}f(x+\frac yb,y), &0<x<1-\frac yb,\\\frac ...
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0answers
14 views

Write the norm $||R^h v||$ just in function of $v$ if $R=[ Q \, \mathbf{0}]$ and the columns of $Q$ form an orthonormal basis

Let $Q$ be an $m \times (m-k)$ (complex) matrix where its columns form an orthonormal basis ($m-k$ vectors). We define matrix $R=[ Q \, \, \mathbf{0}]$, where $\mathbf{0}$ is the $m \times k$ zero ...
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1answer
25 views

Convergence of measures on $\mathbb{T}$

Denote by $M(\mathbb T)$ the set of complex-values measures on the circle $\mathbb{T}=\mathbb{R}/\mathbb{Z}$.Prove that $D(T)$, the set of discrete measures on $\mathbb{T}$ is: closed in ...
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0answers
12 views

comprehensive overview of techniques for finding the difference between vectors

There are many techniques for finding the difference between two vectors. for example: the norm of the difference. absolute value of the difference Mahalanobis distance Bhattacharyya distance etc ...
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1answer
35 views

Interchange of $\ell^r$ and $L^p$-norm

Let $(f_i)_{i\in\mathbb{N}}$ be a sequence of $L^p$-functions. What is the relation between $\Vert \Vert (f_i)_{i\in\mathbb{N}}\Vert_{\ell^r}\Vert_{L^p}$ and $\Vert \left(\Vert ...
4
votes
1answer
31 views

Source/explanation for this matrix inequality

Here it is: $$z^\top M^{-1} M^{-1}z \le \|M^{-1}\| z^\top M^{-1} z.$$ Where $\pmb M$ is positive definite symmetric, $z$ is a vector in $\mathbb{R}^p$ (not necessarily normed!) and $||\pmb M||$ is ...
2
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1answer
28 views

Open mapping theorem for normed abelian groups

A norm on an abelian group is a function valued in $\mathbb{R}_{\geq 0}$ which satisfies $|x|=0 \Leftrightarrow x=0$, $|{-}x|=|x|$, and $|x+y| \leq |x|+|y|$, not necessarily $|z x| = |z| |x|$ for ...
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2answers
35 views

Exchanging limits with norms and linear functionals

In a normed vector space $X$, when can we say: $\lim\|x_n\|=\|\lim x_n\|$ and further, if $f\in X^{*}$, when can we say: $\lim fx_n=f(\lim x_n)$?
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1answer
47 views

Show that this matrix product is bounded

Suppose we have a symmetric real matrix $\pmb M$ satisfying: $$\underset{\pmb\alpha\in\mathbb{R}^p:||\alpha||=1}{\min}\;\pmb \alpha\pmb M\pmb \alpha^{\top}\geqslant k>0.$$ Then, I am trying to ...
2
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1answer
19 views

Proving inequality for norm of linear transformation

Stumbled upon this one in a textbook: Let there be a linear transformation $T:V\rightarrow V$ over a finite inner product space $V$. It is known that $TT^* = 7T - 12I$. How can it be proved that ...
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10 views

How to find the normal of the intersection of 2 vectors on a plane.

I am wondering how I would go about finding the normal of plane when I don't know the equation of said plane. I am drawing a cube so I have the corner points of a square, and I need to find the the ...
2
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1answer
32 views

$l^{2}$ completness in the given norm

Let's consider a $l^{2}$ space, equipped with a norm $$||x||_{\infty} = \sup_{n}{|x_{n}|}+\sum_{n=1}^{\infty}{2^{-n}|x_{n}|}$$ I would like to establsh, whether the space is complete in the given ...
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70 views

linear problem with $\|.\|_\infty$ and $\|.\|_1$ norm constraints

I have a question regarding a straightforward linear algebra problem, yet the solution is (at least for me) not trivial. Assume the sequences $\phi_i$ with coefficients $\phi_i[n]\in\mathbb{R}$, and ...
0
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1answer
19 views

The norm of trace of functions in $H^\frac{1}{2}(\partial\Omega)$

Let $\textbf{A}\in(H^1(\Omega))^3$, where $\Omega\subset\mathbb{R}^3$ is a bounded convex domain with its boundary $\partial\Omega$. Now we know, on $\partial\Omega$, ...
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vote
1answer
30 views

Induced norm of a real matrix, symmetric, positive definite

Describe geometrically $\left\{x\in\mathbb{R}^{n}:\left\Vert x\right\Vert _{A}=1\right\}$ where $\left\Vert \cdot\right\Vert _{A}$ is the induced norm of a real matrix, symmetric, positive definite.
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2answers
56 views

Under what condition does $\int_a^b|f(x)|=0$ imply $f=0$? ( at least a.e.).

I'm working through Real Analysis by Royden & Fitzpatrick, and on the first section on $L^p$ spaces they always skip the last property of norms without much commentary. Namely, that $||f||=0 \iff ...
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1answer
32 views

conjugate transpose of contraction

is it true, for any matrix $T \in \mathbb{C}^{n \times n}$ (with scalar product $\langle .,.\rangle$ and the norm $\left\Vert v\right\Vert = \langle v,v\rangle^{\frac{1}{2}}$), that, if $T$ is a ...
0
votes
2answers
21 views

$\not\exists$ number $c\geq 0$ for which $||f||_{max}\leq c||f||_1$. But $\exists$ $c\geq 0 $ for which $||f||_1\leq c||f||_{max}$

I'm reading through Real Analysis by Royden & Fitzpatrick and I always try to deal with the first few problems of each section. For the first section on $L^p$ spaces there were mostly routine ...
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3answers
33 views

Derivative of matrices product

Find the derivative of the following matrix $ f(X) = a^TXb, $ where $ a,b ∈ R^n $ and X is an n×n matrix. Please give me some serious hint!
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1answer
27 views

If $f_n\to f$ in the $L^1$ norm, show that there is a subsequence $f_{n_k}$ which converges a.e. to $f$.

This question is from a problem set on $L^p$ spaces in my undergraduate-level real analysis course. I said that $f_n$ converges if and only if it is Cauchy. Therefore, $\exists \, N\in\mathbb{N} \; ...
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2answers
30 views

Does $\{\sin (nx)\}_1^\infty$ converge in the $L^1$ norm on $[0,2\pi]$?

This is a homework question from a problem set in an undergraduate-level real analysis course (coming from merely an intro to analysis course) about $L^p$ spaces. Show that $\{\sin ...
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1answer
20 views

Derivates of a vector in respect to the elements

Find the derivative of $(a) f(x)= \frac{1}{x_3} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ $(b) f(x)=trace(xx^T)$ where $ x_1 , x_2, x_3 $ are the first three elements of x.
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1answer
21 views

Jacobian of second norm

Find the Jacobian of the following function: (a) $f(x)= \|x -x_0 \|_2$ (b) $f(x)= \log(\|x \|_2)$ Please give me some serious hint!!
2
votes
1answer
27 views

Derivative of a nonsingular matrix

Show that : $$\frac{d}{dt} A^{-1}(t) = -A^{-1}(t) (\frac{d}{dt} A(t) ) A^{-1}(t) $$ A(t) is a matrix.
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2answers
50 views

Cauchy Sequence in Normed Space

Let $(E, ||\cdot ||)$ be a normed space and let $(x_n)$ be a sequence in $E$. Show that the following conditions are equivalent: (a) $(x_n)$ is a Cauchy sequence. (b) For every increasing function ...
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votes
1answer
13 views

Proof for Semidefinite symmetric matrix product

If A is symmetric positive semidefinite, show that: (a) For any matrix B,$ BAB^T $is also positive semidefinite. (b) All the diagonal elements of A are nonnegative.
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21 views

Minimization using Singular value

Let $A$ be a $p\times q$ matrix, with rank $q$. Show that the vector $x$ that minimizes $\|Ax\|_2$ under the constraint $\|x\|_2 = 1$ is the right singular vector of $A$ corresponding to the smallest ...
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2answers
25 views

How to maximize matrix products

Find a unit vector v1 and a unit vector v2, such that the term: $$v^T \begin{bmatrix} 6 & -2 \\ -2 & 6 \end{bmatrix}v$$ is minimized and maximized, respectively. What are the minimum and ...
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2answers
22 views

Multiplication of an orthogonal matrix by its first column

I am given a real orthogonal matrix Q (nxn), where the first column of Q is the vector x (nx1) where the 2-norm of x equals 1. I am asked to prove that QTx has first entry 1 and all the others zero: ...
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3answers
355 views

Norms and equivalence classes question

Let $f\in C[0,1]$. Recall two of the norms we considered in class: $$\|f\|_\infty = \sup_{t\in[0,1]}|f(t)|, \quad \|f\|_1 = \int_0^1|f(t)|\ \mathsf dt. $$ Consider the space $C^1([0,1])$ of ...
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1answer
27 views

lower bound on the minimum singular value of $\underline{\sigma} (A+B)$

what can we say about the lower bound on $\underline{\sigma}(A+B)$? Can we say the following? $\underline{\sigma}(A+B)>\underline{\sigma}(A)+\bar{\sigma}(B)$, where $\underline{\sigma}$ denotes ...