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

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norm of lower triangular part bounded by norm of matrix

Suppose $A \in \mathbb{C}^{4 \times 4}$ and $L$ is the lower-triangular part of $A$. Show that $||L||_2 \leq 3||A||_2$. Here $||\cdot||_2$ is the spectral norm. I have been given 2 hints: ...
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32 views

Is there a nuclear norm approximation for stochastic gradient descent optimization?

I want to minimize $E$ by using stochastic gradient descent. I know that there is a sub-differential for the nuclear norm, but i want to know if is there a approximation of nuclear norm in order to ...
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50 views

Prove 1-Norm is a Norm

I am just curious how you would simply prove that a 1-norm is a norm. Step-by-step would be very helpful. Proofs are not my strong point. Thank you!
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24 views

Using the definition of the operator norm

I am given the following problem: Using the definition $$\lVert L \rVert_{\text{op}}=\sup_{\vec{u} \in \mathbb{R}^d, \lVert \vec{u} \rVert=1}\lVert L\vec{u} \rVert$$ of the operator norm of a ...
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1answer
53 views

Detailed working of $||f||$

Given $\ell^{\infty} \rightarrow \mathbb R$ defined by $$f(a_1,a_2, a_3, ...)=\frac 1{\sqrt {0!}}a_1 + \frac {-1}{\sqrt {1!}}a_2 + \frac 1{\sqrt {2!}}a_3 +... +\frac {(-1)^{n-1}}{\sqrt {(n-1)!}}a_n$$ ...
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1answer
42 views

Gradient of the Frobenius Norm (or matrix trace) of an expression involving a matrix and its inverse

For real, positive definite (square) matrices $\mathbf{A}$, $\mathbf{X}$, and $\mathbf{C}$, I would like to find an expression for the following gradient: $\nabla_\mathbf{X} || \mathbf{AX}+\mathbf{X}^...
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31 views

definition of norm and $\ell_2$ norm

This is from my textbook: I am pretty sure when we have a norm function for a vector, for example $v=(3,4)$, so $\|v\| = 5$, which is the euclidean distance, so why the '2' is deleted rather than ...
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38 views

$L^2$ and $L^\infty$ normed inequality for PDE solution: Which one is more informative and why?

I have the following inequalities $$max_{t \in [0,T]} \lVert u_1(t, \cdot)-u_2(t, \cdot) \rVert_{L^2(\mathbb{R})} \leq C \lVert g_1(x) - g_2(x) \rVert_{L^2(\mathbb{R})}$$ and $$max_{t \in [0,T]} \...
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1answer
33 views

Spectral Norm Proof

I don't really understand the question below: A matrix can have small eigenvalues but large spectral norm (i.e., largest singular value). Find such a matrix A of proper dimension so that all of its ...
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What does this (double absolute value like) notation mean?

Here, $$\left\lVert\frac{\partial\bf x}{\partial s}\times\frac{\partial\bf x}{\partial t}\right\rVert$$ the inside will at last be a vector. and two absolute value signs have covered it. what does ...
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32 views

Check $F_2(f)=f'(1)+f(1)$ is a linear functional

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

Show $L$ is a closed linear subspace of $H$

Let $H$ be a Hilbert space and linear operator $P:H \rightarrow H$ satisfy $<Px,y>=<x,Py>$ for all $x,y \in H$ and $P^2=P$. We can use the fact that $Px \perp (x-Px)$ for every $x \in H$ ...
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3answers
50 views

Show $Px \perp (x-Px)$ and prove $||P|| \leq 1$

Let $H$ be a Hilbert space and linear operator $P:H \rightarrow H$ satisfy $\langle Px,y\rangle = \langle x,Py\rangle$ for all $x,y \in H$ and $P^2=P$. (a) Show that $Px \perp (x-Px)$ for every $x \...
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33 views

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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1answer
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 x=(a_1,a_2,\...
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1answer
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,...), \, \, \, \, \, x=(a_1,...
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1answer
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,...), \, \, \, \, \, x=(a_1,...
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32 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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3answers
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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46 views

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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28 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, diagonal,...
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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 $$\frac{\|(A+M)^{-1} - A^{-1} \|}{\|A^{-1}\|}\le 2c.$$...
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1answer
144 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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1answer
29 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 $\displaystyle\...
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1answer
25 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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67 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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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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1answer
32 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} & \frac{...
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49 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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1answer
78 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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1answer
131 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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40 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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43 views

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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3answers
72 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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1answer
9 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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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 $\|Tx\|...
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1answer
33 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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1answer
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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35 views

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 norms $\|...
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14 views

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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1answer
48 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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0answers
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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2answers
44 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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1answer
17 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 ...
5
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1answer
858 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 ...