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

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Submultiplicative Matrix Norm

Given $I$ is the $n\times n$ identity matrix, $F$ an $n\times n$ matrix with $\|F\|_{M} < 1$ for some submultiplicative norm $\|.\|_M$, by which formula is $(I+F)^{-1}$ calculable? And is it ...
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finding operator norm $T_N$

How do find the operator norm of , $T_N\colon c_0\to \Bbb R$ given by $T_N(y):=\sum_{j=1}^Nx_jy_j$ ,when $N$ is a integer . TIA
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113 views

Does π depends on the norm? [duplicate]

If we take the definition of π in the form: π is the ratio of a circle's circumference to its diameter. There implicitly assumed that the norm is Euclidian: \begin{equation} ...
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A bound on the norm of the sum of two index-disjoint matrices

Given two matrices, it is well known that $\parallel A+B \parallel _2 \leq \parallel A \parallel _2+\parallel B \parallel_2$. Now, suppose that the nonzero indices are disjoint (i.e., $A$ is nonzero ...
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Invertible operator norm bound

Let $H$ be a Hilbert space and that $X$ are bounded. Suppose $X$ is self-adjoint. Show that $Y=X+iI$ is invertible and the inverse $Y^{-1}$ has the norm $\lVert Y^{-1} \rVert \le 1$. I can prove $Y$ ...
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two topology questions (open set and equivalence)

Two metrics $d_1$ and $d_2$ are called equivalent if there exist positive constants $\alpha, \beta$ s.t $\forall x,y\in\mathbb R^n: \alpha d_2(x,y)\le d_1(x,y)\le\beta d_2(x,y)$ I already proved that ...
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49 views

Does $\|z\|=(\sqrt{|x|}+\sqrt{|y|})^{\frac{1}{2}}$ define a norm on $\mathbb R^2$

Does $\| z\|=(\sqrt{|x|}+\sqrt{|y|})^{\frac{1}{2}}$, with $z=(x,y)\in\mathbb R^2$, define a norm on $\mathbb R^2$?
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Hölder norm estimates

How do you prove the following estimate for composition of functions: If $k\geq 1$, then there exists a constant $c=c(k,\alpha)$ such that $$ \|f_1\circ g_1-f_2\circ g_2\|_{k,\alpha}\leq ...
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Counterexamples of Arzèla Ascoli theorem for non-obeyed criteria

I had an exam on functional analysis some time ago, and one of the questions I couldn't make any sense out was the following: Let $\Omega\subset \mathbb{R}$ and $\{f_n\}$ a sequence of continuous ...
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A specific linear operator between Banach spaces

Let B be the Banach space $B=(C[0,1],\|\cdot\|_{\infty}$) and let $\{\xi_i\}\in l^\infty$. Let $T:l^1\rightarrow B$ be the linear operator given by: $(Ta)(x) = \sum_n\xi_na_nx^n$. I have three ...
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Operator norm converging to 0 for certain condition

Let $X$ be a finite-dimensional normed space and $T_n : X \to X$ a sequence of linear operators such that $\lim_nT_nx = 0$ for all $x$ in $X$. Prove that $\lim_n\|T_n\|=0$.
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The norm of an operator

Let $\rho(x)$ be a weight function in a unit sphere, such that \begin{equation} \begin{array}{l} \displaystyle 1. \rho(x)\ge 0,\int_{\mathbb{R}^n}\rho(x)=1\\ \displaystyle 2. \rho(x)\in ...
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Two proofs concerning Hölder's inequality

I am studying functional analysis and I have come across two statements which can be proven by using Hölder's inequality, but I don't know how/why. Reminder: let $a\in l^p$ and $b\in l^q$, then: $$ ...
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The $ l^{\infty} $-norm is equal to the limit of the $ l^{p} $-norms. [duplicate]

If we are in a sequence space, then the $ l^{p} $-norm of the sequence $ \mathbf{x} = (x_{i})_{i \in \mathbb{N}} $ is $ \displaystyle \left( \sum_{i=1}^{\infty} |x_{i}|^{p} \right)^{1/p} $. The $ ...
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How to show that $\left \| AB \right \|_F \leq \min (\left \| A \right \|_F \left \| B \right \|_2 , \left \| A \right \|_2 \left \| B \right \|_F)$?

For any matrices $A \in \mathbb{C}^{m \times k}$ and $B \in \mathbb{C}^{k \times n}$, show that $\left \| AB \right \|_F \leq \min (\left \| A \right \|_F \left \| B \right \|_2 , \left \| A \right ...
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equivalence of norms

I would like a little help here: I have two defined norms over $C^{1}([0,1])$ : $\| A(f)\|=|f(0)|+\max_{x\in[0,1]}{|f'(x)|}$ $\| B(f)\|=\int_0^1|f(x)|dx+\max_{x\in[0,1]}{|f'(x)|}$ I already ...
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Optimizing over norms of set of equations.

I have a set of N real-valued finite-dimensional vectors $\mathbf{v}_i$ and target norms $y_i$ and I am trying to find a linear transformation matrix $L$ such that the norm of the transformed vectors ...
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311 views

norm equivalence

Exercise 7. of Terence Tao's blog on random matrix, specifically on eigenvalues of Hermitian matrices ...
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Operator norm and spectrum

Let $L$ be a bounded linear operator on a Hilbert space. Without assuming finite dimensions, can we express the operator norm of $L$ in terms of the spectrum of the positive operator $L^{\dagger}L$? ...
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About the convexity of Ky Fan's norm

As we know, the Ky Fan norm is convex, and so is the Ky Fan k-norm. My question is, does this imply that the difference between them is a non-convex function, since it results from "difference between ...
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Examples of functions that are Lipschitz w.r.t. Schatten p-norm?

A convex function $f$ is $R$-Lipschitz w.r.t. to a norm $\|\cdot\|$ if for all points $a, b$ we have $|f(a)-f(b)| \leq R\|a-b\|$. For a real symmetric $n\times n$ matrix $A$ with eigenvalues denoted ...
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On the convexity of element-wise norm 1 of the inverse

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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Minimum of Frobenius norm

I'm not sure if this solvable. I would like to find an orthogonal matrix that minimizes the following Frobenius norm: $$\min||BQ||^2 \\ \text{s.t. } Q^TQ=diag(\alpha_1,...,\alpha_n)$$ where ...
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Difference between orthogonal and orthonormal matrices

Let $Q$ be an $N \times N$ unitary matrix (it's columns are orthonormal). I understand since Q is unitary, it would preserve the norm of any vector $X$, i.e, $||QX||^2=||X||^2$. My confusion comes ...
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Norm equivalence Sobolev space

I have this problem: Let $k>0$ (integer) and $1 \leq p < \infty$. Show that the norms $$ ||u||_{W^{k,p}(U)} = \bigg( \sum_{|\alpha|\leq k}||D^{\alpha}u||_{L^{p}(U)}^{p}\bigg)^{\frac{1}{p}} $$ ...
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Sobolev spaces doubt

Can somebody help me with this doubt? Let $\Omega$ an open set and $A$ be any finite subset of points of $\Omega.$ Is it true the following inequality? $\vert v(a) \vert \leq C \| v \|_p ...
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Matrix norms involving singular values

The induced 2-norm of a $m \times n$ matrix $A$ is known as $$ ||A||_2 = \underset{||x||=1}{\text{max}} ||Ax||_2 = \sigma_1,$$ where $\sigma_1$ is the largest singular value of $A$. Then, is it ...
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Lower bound on the norm of product of non square matrices

The following inequality is known: $\parallel AB\parallel\geq\parallel A\parallel \sigma_{n}(B)$. However, it is only valid where both $A$ and $B$ are square. Is there an analogue for rectangular ...
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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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How to find the norm of this bounded linear functional?

Let $C^\prime[a,b]$ denote the normed space of all continuously differentiable real (or complex) valued functions defined on the closed, bounded interval $[a,b]$ in $\mathbf{R}$ with the norm defined ...
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Which of the following functions are norms? [duplicate]

For $x=(x_1,x_2)$, which of the following functions on $\mathbb{R}^2$ are norms? a.) $A_1(x) = 7\mid x_1\mid + 3\mid x_2\mid$, b.) $A_2(x) = \text{max}\lbrace\mid x_1\mid^2,\mid x_2\mid^2\rbrace$, ...
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Does a norm have to map to $\mathbb R$?

Wikipedia defines a norm on a vector space $V$ as a function $p : V \mapsto \mathbb R$. I've seen this defined similarly elsewhere. However, it seems to me that a real codomain isn't always necessary. ...
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An inequality involving the norms of symmetric positive definite matrices

Given A and B two real symmetric positive definite matrices is it true that, for some norm $\|.\|$, this inequality holds $$ \|AB-I\| \leq \|A^2B^2-I\| \qquad ? $$
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Analysis.. Norm on C([a,b])

Let $w:[a,b]\rightarrow \mathbb{R}$ with $ w(x)\geq c>0 $ for some $c \in \mathbb{R}$ and all $x \in [a,b]$. Prove that $$\lVert f\rVert_w \ = \ \displaystyle\int^b_a \lvert f(t)\rvert w(t)\ ...
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Analysis.. Convergence of sequence

I really struggle with understanding convergence and have the following questions.. Determine whether the following sequences converge and if so, give the limit: $x_n = ...
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Norms Abstract Analysis

I have a question relating to norms and have been giving functions and need to state whether they are norms or not... which of the following are norms on $\mathbb{R}^2$? Give reasons for your ...
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Help me understand this proof (showing that something is a norm).

I am reading the following paper: Takáč, Peter On the Fredholm alternative for the p-Laplacian at the first eigenvalue. Indiana Univ. Math. J. 51 (2002), no. 1, 187–237. I need help to understand the ...
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Norm of differentiation operator $Tf(t)=f^{'}$..

Consider $T:C^1[0, 1]\rightarrow C[0, 1]$ given by $Tf=f'$ where $$\|f\|_{C^1}=\|f\|_\infty+\|f'\|_\infty$$ and $\|f\|_\infty=\sup_{x\in [0, 1]}|f(x)|$. How to prove $\|T\|=1$? The inequality ...
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Is the polar set of convex Polytope also Polytope

Let $P$ be a convex polytope. How can I prove that the polar set of $P$ (lets call it $P^*$) is polytope? where $P^*=\{x\in\mathcal R^n:\forall v\in P, |\langle x,v\rangle|\le1\}$ . $Thanks$
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Change in singular values of matrix after left-multiply with a diagonal matrix

Say that we have an SVD for a matrix $X = U \Sigma V^T$, giving trace norm $||X||_{tr} = ||\Sigma||_{tr} = \sum \Sigma_{ii}$. I am wondering what happens to the SVD and/or trace norm if we left ...
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Norm to evaluate precision

I have two vectors x and y with their respective n coordinates/components and this norm between them: $$ norm = \sqrt{\frac{\displaystyle\sum_{i=1}^{n} (x_{i}-y_{i})^2}{\displaystyle\sum_{i=1}^{n} ...
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How to show the unit ball of the dual norm is also polytope?

Assuming the norm's unit ball is a convex polytope. How can one show that the unit ball of the dual's norm is convex polytope and/or polytope ?
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On the norm of a quotient of a Banach space.

Let $E$ be a Banach space and $F$ a closed subspace. It is well known that the quotient space $E/F$ is also a Banach space with respect to the norm $$ \left\Vert x+F\right\Vert_{E/F}=\inf\{\left\Vert ...
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Step functions dense in Integrable functions with respect to $L_2$

Let $I$ be a bounded interval. Prove that $\{\text{step functions }I \to C\}$ is dense in $\{\text{integrable functions }I \to C\}$ (Riemann Integrable) with respect to $\|.\|_2$ ($L_2$ norm)
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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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Exercise from textbook about norm

The below diagram included a relevant page and the question 4 of the exercise which i am not sure how to do. Any hint or solution are welcome Attempts: a) I have done it b) i have tried to show that ...
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A limit superior question in the context of the Neumann series

I'm trying to understand a step in the proof that the Neumann series converges: Let $X$ be a Banach space and $T\in L(X)$ (the space of bounded, i.e. continuous linear maps $X\to X$). It is known ...
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Approximating asymmetric matrix's Ky Fan norm

Given an N-by-N asymmetric matrix $M$ Is there any theory about approximating $M$'s Ky Fan k-norm $|| M ||_k$ using $\frac {(M+M^T)}{2}$'s Ky Fan k-norm $|| \frac {(M+M^T)}{2} ||_k$? UPDATE: ...
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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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Norms of idempotent matrices

So if I have a random idempotent matrix (a matrix that when squared equals itself) how do I go about calculating its 2-norm? I know that a idempotent matrix has eigenvalues of only 0 or 1 and I know ...