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

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Operator norm of matrix

Consider $\mathbb{R}^2$ with norm $\Vert (x,y) \Vert=\sqrt{x^2+y^2}.$ I would like to compute the operator norm w.r.t. the above norm of a matrix $$A=\begin{pmatrix} a_{11} & a_{12} \\ a_{21} &...
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62 views

Prove this integral inequality

Prove this assuming $f$ is integratable: $$\int_{-\pi}^\pi\vert f(t)\vert \, dt\leq \sqrt{2\pi}\sqrt{\int_{-\pi}^\pi\vert f(t)\vert^2}\, dt =2\pi \Vert f\Vert.$$ I tried to square both sides and use ...
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58 views

Why is this vector of unit length?

Given $A$ a symmetric positive definite real matrix and $V$ a non zero column vector define: $B=V^\top A^{-1} V$ and since $A$ is p.d., $B>0$. Also, $A$ has a unique square-root $\sqrt{A}$. I ...
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1answer
23 views

the norm a functional that maps a convergent sequence to its limit

Given the linear space $\vec{E}$ of real convergent sequences, dotated with the norm \begin{equation*} \begin{aligned} \| \cdot \| : \vec{E}&\to \mathbb{R} \\ x=(x_n)_{n\in\mathbb{N}} &\...
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27 views

Prove an inequality involving a norm

We define the following inner product on intergrable, $2\pi$ periodic functions from $\mathbb{R}$ to $\mathbb{C}$: $$\langle f,g\rangle = \frac{1}{2\pi} \int_{-\pi}^\pi f(t)\overline{g(t)}\ dt$$ I ...
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80 views

If $V$ is a finite dimensional with two norms then $\Vert v\Vert_1 \leq c\Vert v\Vert_2 $

Suppose $V$ is finite-dimensional and $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$ are inner products on $V$ with corresponding norms $\Vert \cdot\Vert_1$ and $\Vert \cdot\Vert_2$. ...
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1answer
46 views

Hilbert space sequence

Let $$ L^2:=\left\{f:[0,1] \to \mathbb R \; \middle| \; \int_0^1 |f|^2 < \infty\right\}, $$ $$ \langle f,g \rangle:= \int_0^1 f g, $$ and $$ \|f\| := \sqrt{\langle f, f \rangle}. $$ Prove that $\|...
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1answer
59 views

Prove the squared vector 2-norm is $\leq$ sum of 1-norm and infinity-norm

How do I prove that $$\|x\|_2^2 \leq \|x\|_1 \|x\|_\infty?$$
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29 views

Finding the norm of the sum of two vectors

So, there's this thing called "dipole moment" ($\vec{\mu}$) in chemistry. It's basically a vector. An example of that would be a water molecule: dipole moment of $\mathrm{H_2O}\ (\vec{\mu_{\mathrm{...
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77 views

On volume forms and norms on exterior powers

Let $V$ be a $2$-dimensional vector space. Given an inner product on $V$ one may define an inner product on the simple $k$-vectors of $\Lambda^k(V)$ by $$\langle v_1 \wedge \cdots \wedge v_k, w_1 \...
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33 views

Sequence of functions $\sin^{n} (\pi x)$ cauchy in C[0,1]

Equipping C[0,1] with its usual (supremum) norm, for each $n \in \mathbb{N}$, let $g_n(x)=\sin^{n} (\pi x)$ for $x \in [0,1]$. I'm trying to prove (or disprove) this sequence is cauchy. Intuitively, ...
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1answer
62 views

Condition number of 2x2 nonsingular matrix

I am working on the problem that I have to show why the infinity-norm condition number and 1-norm condition number of 2x2 nonsingular matrix are equal. MY ATTEMPT: Since $I=AA^{-1}$, the condition ...
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1answer
44 views

How can I prove that the norm of the following operator is $\frac{1}{m!}$?

I'm trying to prove that the norm of the multilinear symmetric operator $A$ is $\frac{1}{m!}$ where $A$ is defined as: $$ A(x_1,\dots, x_m) = \frac{1}{m!} \sum_{\sigma \in S_m} \xi_1(x_{\sigma(1)} ) \...
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2answers
44 views

An inequality related to matrix norm, inverse matrix

The question is that: We have two matrices $A,B\in \mathbf{C}^{n\times n}$, A is nonsingular and B is singular, let $||\cdot ||$ be $\textbf{any}$ matrix norm, prove $||A-B||\geq 1/||A^{-1}||$. My ...
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$X:\Vert X\Vert_2<1 \iff\text{ matrix }\begin{bmatrix} I&X^*\\X&I\\\end{bmatrix} $ is positive

Following question seems so simple, yet I could not come up with a solution. I started to think that there might be sth wrong with the question. Could you please take a look? For a matrix $X:\Vert X\...
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3answers
101 views

Find the norm of the following operator.

Take $X=C([0,1])$ with the uniform norm, $\|f\|=\sup_{x\in[0,1]}|f(x)|$, and define the operator $T:X\to X$ by, $$T(f)(x)=f(x)-\int_0^1f(s)ds$$ Find $\|T\|$. I was hoping to solve this problem ...
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39 views

Randomly select vector with bounded norm

I would like to select randomly an $n$-dimensional real vector $\mathbf{x}=(x_1,\ldots,x_n)^\top\in\Bbb{R}^n$ such that its norm is bounded by a positive real number, say $\sqrt{a}$, $a\in\Bbb{R}_+$. ...
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3answers
52 views

Orthogonal Projection - property of an orthogonal operator or something that needs to be proven?

I am currently working on showing that $\Vert Px \Vert_{2} \leq \Vert x \Vert_{2}$, where $x \in$ an inner product space $X$, and $P$ is the orthogonal projection operator. Also, I am supposing that $...
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1answer
38 views

Prove that Prove $\left\lvert \left\lVert x \right\rVert - \left\lVert y \right\rVert \right\rvert \le \left\lVert x-y \right\rVert$

This is Exercise 3.1.4 from Economic Dynamics, Theory and Computation by John Stachursky. Key definitions for the exercise are his definition of norm and metric I believe. Let Prove $\left\lVert \...
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1answer
58 views

Showing that the operator is bounded and find its norm.

I have this operator $T: L^p(0,\infty)\rightarrow L^p(0,\infty)$, $1<p<\infty$ : $(Tf)(x)=1/x\int_0^xf(t)dt$. I am supposed to show that it is bounded and fint its norm. I had an idea that ...
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1answer
22 views

Norm Product expression

Prove the product expression $$\left \| AB \right \|_{U\rightarrow W} \leq \left \| A \right \|_{V\rightarrow W}\left \| B \right \|_{U\rightarrow V}$$ Hint: consider $(AB)u = A(Bu)$ and apply $\left ...
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118 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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0answers
32 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 $\mathbb{T}$. I ...
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28 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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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 ...
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1answer
20 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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25 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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40 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
36 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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31 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
74 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
53 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} u(t)v(...
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10 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
70 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 $...
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1answer
40 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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34 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 S_{||\cdot||_{...
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1answer
45 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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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 ...
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1answer
81 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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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
22 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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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
27 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 $M(\...
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1answer
17 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
43 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 f_i\Vert_{L^p}\right)_{...
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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
33 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
38 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
48 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
29 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 ...