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

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Is this function a norm?

Let $p$ be a real number such that $p \geq 1$, let $a$ and $b$ be real numbers such that $a < b$, and let $X$ be the set of all the real- (or complex-) valued functions that are defined and ...
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33 views

Find $\|T\|_{\ell^\infty \rightarrow \ell^2}$ [duplicate]

Let $(v_n)_{n \geq 1} \in \ell ^2$ be a fixed bounded sequence of real numbers. Define a mapping $T$ on $\ell^\infty$ using the formula $$T(a_1, a_2,\ldots)=(v_1a_1,v_2a_2,\ldots), \qquad ...
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18 views

Show $||Tx||_2 \leq ||(v_n)||_2 \cdot ||x||_{\infty}$

Let $(v_n)_{n \geq 1} \in \ell ^2$ be a fixed bounded sequence of real numbers. Define a mapping $T$ on $\ell ^{\infty}$ using the formula $$T(a_1, a_2,...)=(v_1a_1,v_2a_2,...), \, \, \, \, \, ...
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21 views

Show $Tx \in \ell ^2$ for every $x \in \ell ^{\infty}$

Let $(v_n)_{n \geq 1} \in \ell ^2$ be a fixed bounded sequence of real numbers. Define a mapping $T$ on $\ell ^{\infty}$ using the formula $$T(a_1, a_2,...)=(v_1a_1,v_2a_2,...), \, \, \, \, \, ...
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31 views

Show $F_1$ is a continuous linear functional

Let $(X,||\cdot ||)=(C[0,1],||\cdot ||_{\infty})$ and $F_1 :X \rightarrow \mathbb R$ be defined by $$F_1 (f)=\int _{1/2}^{3/4} f(t) dt$$ Show $F_1$ is a continuous linear functional. So we need to ...
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31 views

Norm of a functional is a norm on a v.s. $X^{*}$

Prove that $\| \cdot \|_{X^{*}}$ is indeed a norm on $X^{*}$, the space of bounded linear functionals on a normed space $(X, \| \cdot \| )$. I am not sure what to do in this. I do know that we ...
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Linear functional: continuous at $x_0=0 \iff$ continuous at all $x$

Let $f$ be a linear functional on a normed space $(X, ||\cdot ||)$. Prove that $f$ is continuous $\iff$ it is continuous at all $x \in X$. Backward direction is trivial: Since $f$ is continuous at ...
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25 views

Relationship between matrix structure and size of matrix norms

I am attempting to compute a smallest norm of a matrix (it can be any norm). Are there any results that allow one to say which norms are smaller for certain classes of matrices (for example, ...
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20 views

How to show this inequality with matrix norms?

Let $A$ be invertible and $M$ of the same size as $A$. Show that if $$\frac{\|M\|}{\|A^{-1}\|}\le c \le \frac{1}{2}$$ then $A+M$ is invertible and $$\frac{\|(A+M)^{-1} - A^{-1} \|}{\|A^{-1}\|}\le ...
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141 views

log norm inequality for lower triangular part of matrix

Suppose $L$ is the lower triangular part of a matrix $A \in \mathbb{C}^{n\times n}$. Prove that $||L||_2 \leq ||A||_2 \log_2(2n)$. Here $||\cdot||_2$ is the matrix norm induced by the $p=2$ vector ...
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27 views

$\sup_{x \in \mathbb{R}, x \neq 0} \frac{\|Ax\|}{\|x\|}$ equivalent to $\sup_{\|x\| = 1} \|Ax\|$ [duplicate]

I have seen it mentioned in many places that for some matrix $A \in \mathbb{R}^{n \times n}$ $\displaystyle\sup_{x \in \mathbb{R}^n, x \neq 0} \frac{\|Ax\|}{\|x\|}$ is equivalent to ...
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21 views

Inner Product on Sobolev Space with p=2

Wikipedia defines the Sobolev Space: $H^{s,p}(\mathbb{R}^n)= \left\{f \in L^p(\mathbb{R}^n): \mathcal{F}^{-1}[(1+|k|^2)^{\frac{s}{2}} \mathcal{F}f] \in L^p(\mathbb{R}^n) \right\}$ Where $s \in ...
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62 views

The output of Gram - Schmidt process without normalizing

What happens if I don't do normalization during Gram - Schmidt algorithm, do I still get orthogonal vectors?
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12 views

Inequality with $W^{1,p}$ norm

For $a<b \in \mathbb{R}$, let $G = (a,b)$. How can I show that firstly for every $v \in W^{1,p}(G)$ there exists a unique $\tilde{v} \in C^0(\overline{G})$ such that for almost every $x \in G$ it ...
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9 views

Inequality with $W^{1,p}$-norm

How can I show that for every $p \in [1,\infty)$ and every $v \in C^1_c(\mathbb{R})$ that $$ \lvert g_p(v(x)) \rvert \leq p \lVert v \rVert^p_{W^{1,p}(\mathbb{R})}$$ for every $x \in \mathbb{R}$ where ...
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31 views

Verify that $\| T(x) \| = \| x \|$

Let $T:\mathbb{R}^3 \rightarrow \mathbb{R}^3$ be multiplication by the matrix $$A= \begin{bmatrix} \frac{1}{3} & \frac{2}{3} & \frac{2}{3} \\[0.3em] \frac{2}{3} & -\frac{2}{3} & ...
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39 views

Compare two matrices

Say, I have two matrices $A_1$ and $A_2$, what is a good way to check how similar they are... There have been questions asked before, and the answer suggest Froberius norm, for example. But Frobenium ...
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71 views

Sobolev spaces of sections of vector bundles

Suppose that $X$ is a compact (smooth) $n$-manifold and $E \to X$ be a rank $N$ smooth (complex) vector bundle. Choose finite covering $(U_i)_i$ by domains of the charts $\varphi:U_i \to V_i \subset ...
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124 views

Derivative of Frobenius norm of matrix logarithm with respect to scalar

I am stuck on finding $t$ such that: $\frac{\partial}{\partial t}\|\log_m(M\Lambda^tM^T)\|_F=0$, where $M$ is $n\times n$ positive definite matrix (not symmetric, not unitary), $\Lambda$ is $n\times ...
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39 views

Characterize a norm in $\mathbb{R}^n$

For $\mathbb{R}^n$ we have the classical $p$-norms: $\|x\|_p=(x_1^p+...+x_n^p)^{1/p}$ and $\|x\|_\infty=\max|x_i|$. But... 1) There is another norm in $\mathbb{R}^n$ different for the above ones. 2) ...
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Proving a trivial bound on $L_2$ norm of the error in a sparse approximation of a vector

Trying to understand this supposedly 'trivial' bound from a paper: If $\theta_N$ denotes the vector $\theta$ with everything except $N$ largest coefficients set to $0$ then we have $$ || \theta - ...
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67 views

Question on upper triangular matrix with complex eigenvalues with modulus less than 1

This is problem 16, Section 6.B from Linear Algebra Done Right, 3rd Edition. Suppose the field is $\mathbb{C}$, $V$ is finite-dimensional, $T \in \mathcal{L}(V)$, all the eigenvalues of $T$ have ...
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8 views

A list is a basis if norm of difference with orthonormal basis is bounded by given constant less than unity

This is a problem from Linear Algebra Done Right, 3rd Edition, problem 14 in 6.B. Given $\{e_1, \dotsc, e_n\}$ is an orthonormal basis of $V$, and $||v_j -e_j|| < \frac{1}{\sqrt{n}} \; \forall \; ...
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28 views

Closedness and continuity in infinite dimensional spaces

I cannot understand why the operator $A=d/dx: D(A)(\subset C[a,b])\to C[a,b]$ is closed when the domain $D(A)$ is chosen to be $C^1[a,b]$ while we know that we can converge to a non-differentiable ...
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29 views

Lipschitz constant of a vector valued function

I want to find the Lipschitz constant for $f:\mathbb{R}_{+}^{N}\rightarrow\mathbb[0,1]^{N}$, $$ f_{i}(x)=x_{i}\wedge\left[1-\sum_{j=1}^{i-1}x_{j}\right]^{+},i=1,2,\ldots,N, $$ ($a\wedge b=\min(a,b)$ ...
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Finding two adjoints, and showing boundedness of operators

Let $H = l_2$ and consider the following operators: $T,S:H \to H$ $Tx = (0,x_1,x_2,\ldots)$ and $Sx = (x_2,x_3,x_4,\ldots)$ Show they are bounded, and find the adjoint of both: For $T$, I have ...
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32 views

Bounded distance of powers of a matrix from identity (Frobenius norm) implies that the matrix is identity

Let $A=\left( { a_{i,j} }\right)$ be an invertible $ n \times n$ matrix over $ \mathbb{C} $. Denote $ \|A\|^2 := \sum_{i,j} |a_{i,j}|^2 $ the Frobenius norm. Suppose that for all integer $ k $ we ...
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48 views

showing $\|x\| \leq \lim_{n\to \infty} \inf \|x_n\|$

showing $$\|x\| \leq \lim_{n\to \infty} \inf \|x_n\|$$ where $x_n \to x$ weakly, and we are working under a normed space. I am given a hint that $$\|x\| = \sup_{\|\phi\| = 1} |\phi(x)|$$ where $\phi ...
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30 views

Quadratic form $x^T A x $ is less than $||A|| x^T x$?

Does the following hold? $$x^T A x \leq ||A|| x^T x$$ $A$ is a symmetric positive-definite real-valued matrix, and $x$ is a real-valued vector. Norm is Euclidean.
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If two norms are not equivalent, then we have a sequence $\|x_n\|/\|x_n\|' \to 0$.

Definition 2.22. We say that two norms $\|\cdot\|$ and $\|\cdot\|'$ are equivalent if there exist $C_1,C_2>0$ such that $$C_1\|x_1\|' \le \|x\| \le C_2 \|x\|',$$ for all $x\in V$. If two ...
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Limit Definition for Norms

Let $X_n\in \mathcal{M}_{n\times n}(\mathbb{R})$ be defined as an iterative sequence, $B\in \mathcal{M}_{n\times n}(\mathbb{R})$ and $\|\cdot\|$ be an operator norm. If we are given that $\|X_n-B\|\to ...
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46 views

Does the norm of the product give information about the norm of the matrices?

Let $\|\cdot\|$ be an operator norm subordinate to $\|\cdot\|_{\infty}$ and $A,B\in \mathcal{M}_{n\times n}(\mathbb{R})$. Also, let us assume that $\|AB\|\to 0$. Now, by the multiplicative inequality ...
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49 views

Prove Norm Theorems

I have the following as given: Let $A \in C^{m\times m}$. Then: 1) $$\lVert A\rVert_1 =\sup_{v\in C^m \setminus\{0\} }{\lVert A_v\rVert_1 \over \lVert v\rVert_1} = \max_{j} \sum_i |a_{ij}|$$ How can ...
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43 views

Proving that $F$ is a contraction for a value $\alpha$

Given the differential equation $dy/dx = f(x,y)$ with initial condition $y(x_0)=y_0$. Let $f$ be a continuous function in $x$ and $y$ and Lipschitz-continuous in $y$ with Lipschitz constant L. Let F ...
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1answer
38 views

Normed space related question

Let $p$ be in the range of $0<p<1$, and consider the space $ L_p[0,1]$ of all functions with $$ \|x\| = \left[\int_{i=0}^1 |x(t)^p| \, dt\right]^{1/p} <\infty$$
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16 views

Show that, for a real matrix, its norm is the same when it's seen as a complex matrix

If $A$ is real matrix of size n, and if we take the euclidian norms and the matricial norms associated, how can one show that the norm of $A $ as seen as a complex matrix is the same norm as the one ...
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660 views

A matrix norm inequality

Given a real $m\times n$ matrix $C$, a $m\times m$ diagonal matrix $p$ whose diagonal entries $p_{ii}$ are either 0 or 1, and a $n\times n$ diagonal matrix $q$ whose diagonal entries $q_{ii}$ are ...
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30 views

Reverse triangle inequality in a normed linear space [duplicate]

I need to prove the following equation $$\lvert\lVert x\rVert-\lVert y\rVert\rvert \le \lVert x-y\rVert$$ How can I prove that? I used triangle inequality. But, get stuck.
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19 views

Induced matrix p-norm example

I'm struggling to understand induced matrix p-norms. Our lecture notes have used the example: $$A := \begin{bmatrix}\frac{3}{2} & 0 & \frac{1}{2}\\0 & 3 & 0\\ \frac{1}{2} & 0 & ...
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64 views

Proof of a p-norm and Cauchy/Cauchy-Schwarz inequality generalization

I've started to study metric spaces recently and got intrigued with a demonstration which led me to a bunch of ideas and questions. First of all, I've proved the Cauchy inequality and then ...
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Components of a bounded vector must be bounded

Let $X$ be a vector space over $\mathbb{R}$ with basis $\{x_1, \dots, x_n\}$. Let $c = (c_1, \dots, c_n) \in \mathbb{R}^n$ and suppose $\| c_1 x_1 + \dots + c_n x_n \| = 1$. I'm trying to show that ...
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20 views

Properties of a norm 3

I know that to define a norm, four properties need to be true: norm(x)>=0, norm(x)=0 if and only if x=0, norm(ax) = |a| norm(x) with 'a' a scalar, norm(x+y)<=norm(x)+norm(y). Our teacher mentioned ...
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44 views

Equivalence of complete norms

Let $\|.\|_1$ and $\|.\|_2$ be two complete norms on a linear space $X$ such that if a sequence $(x_n)$ converges to $x$ in $(X,\|.\|_1)$ and to $y$ in $(X,\|.\|_2)$, then $x=y$. We have to prove that ...
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66 views

Prove that a dual norm is of the same form as the norm in original vector space.

Let $\| \cdot \|$ be any norm on $E^n$. Define on $(E^n)^*$, the dual norm as follows. $$\| a \|^* = \max \{ a \cdot x: \lvert \lvert x \rvert \rvert = 1\}.$$ For every covector $a$, (a) Verify ...
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45 views

Poincaré's Inequality on Sobolev Spaces in One Dimension

The following is a version of Poincaré's inequality: Let $I$ be a bounded interval, then there exists a constant $C$ dependent on $I$ such that $$\|u\|_{W^{1,p}(I)} \leq C\|u'\|_{L^p(I)} \ \ \ \ ...
3
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1answer
30 views

Uniform lower bound of sequence of linear maps on a Banach space

Suppose a sequence of $T_j:X\to\mathbb{R}$, with $X$ Banach, has the following property: $$\forall j:\|T_j\|\geq c>0$$ Then we have for all $n$, $$\exists x\in X\setminus \{0\}:\forall j\leq ...
2
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1answer
29 views

Norm operators bounded below implies almost uniform lower bound

I have a hard time proving (or disproving) the following statement about continuous linear operators: $$(\exists c>0:\forall j:\|T_j\|\geq c)\Rightarrow(\exists\delta>0:\forall n:\exists x\in ...
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11 views

Need help with this proof related to semi norms and convergence

I am trying to understand the following proof as shown in the figure. Can somebody help me how (a) and (b) have been proved. The next part of the proof is here.
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1answer
56 views

Prove that there exists NO norm such that $ f_n $ converges to $f$ iff $f_n$ converges to $f$ on compacta.

More specifically: Given the space of continuous functions $\mathbb{R}\rightarrow\mathbb{R}$ where convergence is equivalent to uniform convergence on compacta (i.e., compact sets), prove that ...
2
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1answer
59 views

the greater of two random variables

For two independent normal random variables with non-zero mean $$ X \sim N(u_1,\sigma_1) , Y \sim N(u_2,\sigma_2) $$ If we have the condition, $$E(X^2 ) > E(Y^2)$$ is this condition always ...