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

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Find $\|B\|_{\ell^1 \rightarrow \ell^1} $

$B(a_1,a_2,...)=(a_1/1,a_2/2,\ldots, a_n/n,\ldots) $ This is a linear operator defined on $\ell^1$. Notice that $\|Bx\|=\sum |a_n/n| \le \sum |a_n| =\|x\|$. So $\|B\|=\sup_{\|x\|_1 \le 1} ...
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63 views

Does proper contraction on Hilbert space necessarily lead to convergence in norm to zero?

I was asked this in functional analysis class: Let $ \mathbb{H} $ be a Hilbert space and we are given an operator T satisfying: $ || Th || < ||h|| $ for all $ h \in H $. We are asked if ...
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1answer
17 views

Ratio of eigenvalues as the condition number of a matrix

I'm having an issue with a 2x2 matrix. My understanding is that one could use the ratio of the maximum eigenvalue to the minimum eigenvalue of a matrix in order to determine the condition number, ...
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19 views

Does strict contraction lead to convergence to zero in norm?

In my functional analysis class I was asked this question which got me stuck: Let $ \mathbb{H} $ be a Hilbert space and we are given an operator T satisfying: $ || T || < 1 $ in the ...
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1answer
15 views

Vector difference norm bound of $\|\frac{x_1}{\|x_1\|^2} - \frac{x_2}{\|x_2\|^2}\|$

Given $\|x_1 - x_2 \| \leq C$ where C is a constant, could we derive a bound of $\|\frac{x_1}{\|x_1\|^2} - \frac{x_2}{\|x_2\|^2}\|$?
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1answer
26 views

lim r(t) = L implies lim ||r(t)|| = ||L||

The question is: if $\lim_{t\rightarrow t_0}\vec r(t) = \vec L$ show that $\lim_{t\rightarrow t_0} \|\vec r(t)\| = \|\vec L\|$ So here's where I am so far: let $\vec r(t) = (f_1, f_2,...f_n)$ be ...
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1answer
30 views

Showing that $\langle T(u),T(v)\rangle = \langle u, v \rangle$ implies $T$ is a linear isometry

Let $T$ belong to $\mathcal{L}(H)$ (i.e., the set of linear operators from $H \mapsto H$ where $H$ is a Hilbert space). I need to show that $T$ is an isometry iff $\langle T(u),T(v) \rangle = \langle ...
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1answer
14 views

Norm induced by Dot Product?

Prove, that in $[a,b] \to \mathbb{R}$ bounded and closed interval, the continuously differentiable functions(with complex values) on the set $C^1[a,b]$, $$||f||=(\int_a^b |f(x)|^2 dx + \int_a^b ...
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29 views

How can I prove $||A||_1=$ max$_{1\le j \le n}$ $\sum_{i=1} ^n$ $|A_{ij}|$ ??

How can I prove $||A||_1=$ max$_{1\le j \le n}$ $\sum_{i=1} ^n$ $|A_{ij}|$ ?? $A \in R^{n\times n}$ , $x\in R^n$ The definition of 1 - norm of matrix is: $||A||_1$ = max$_{x\ne 0}$ ...
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18 views

Propagation of Errors, Variance of a $N$ dimensional vector

I am working on a calculation and I'm trying to see if the last step is true. I tried to simplify all the details of the problem. Let $\mathbf{W}$ be a $N\times N$ matrix, $a=\{a_{1,}a_{2},a_{3}\}$ ...
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1answer
36 views

Questions on proof that $\Vert \cdot \Vert_p$ is a norm when dealing with $L^p$ spaces

Unfortunately I never had an opportunity to take a functional analysis/PDE module so I keep running into steps I'm unsure of when I'm looking at proofs. I'm trying to fix that now with some books so ...
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1answer
33 views

Intuitive explanation of p-norm in finite and infinite dimensinos

I am not a mathematician, so very rigorous treatment with things that only a math major learns will not suffice here. I want to learn about p-norms and i can't quite get the intuition behind them. I ...
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44 views

Is $(1,0,…,0,-n^2,0,0,…) \in \ell^2$?

I am a bit unsure of this because if $n$ is very large then the sum would not be finite but then again the term after the nth term is $0$ onwards.
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1answer
28 views

Norm of a matrix formed using a unitary matrix

Suppose, $ A $ is a unitary matrix in $ M_n(\mathbb{C}) $ given by $ (a_{i,j})_{1\le i,j\le n} $ which has the property that, for all the basis elements $ e_i $, $ Ae_i\ne \lambda e_j $ for all $i,j $ ...
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28 views

Vector Euclidean norm upper bound by his coordinates average.

I'm trying to extend the Rademacher complexity and have the following question: For $ (v_1,..,v_m) = V \in {\mathbb{R}}^{m} $ , I will be glad to find an upper abound for the Euclidean norm: $$ ...
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1answer
39 views

Show $\langle f,g\rangle$ is not an inner product

Let $X = C[−1,1]$ be the space of continuous functions $f : [−1,1] → \mathbb R$. For $f,g ∈ X$ define $$\langle f,g\rangle =\int_0^1 f(t)g(t)dt$$ If I choose $f(t)=-t$ and $g(t)=1$, then $\langle ...
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23 views

Solution of Volterra equation in $L_\infty$?

I'm having some trouble proving the following (if it is at all true?). Consider a time-varying Volterra equation $$ F(x, \xi) = f(x, \xi) + \int_\xi^x G(x, s) F(s, \xi) ds $$ for some (known) ...
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101 views

Does $ \int_a^b |f(x) - f_1(x)| = 0$ imply $ \int_a^b |f(x) - f_1(x)|^2 = 0$?

Context:I'm trying to solve this problem: Suppose $f, f_1, g, g_1$ all Riemann integrable complex valued functions on $[a, b]$ such that $f \sim f_1$ and $g \sim g_1$. Prove $\langle f, g \rangle ...
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1answer
33 views

Is the result true when the valuation is trivial and $\dim(X)=n$?

Here I proved the following result: Proposition: Let $(K,|\cdot|)$ be a valued field with non-trivial valuation and let $X$ be a vector space over $(K,|\cdot|)$. Two norms $p_1,p_2$ on $X$ are ...
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40 views

Are there finite dimensional $\mathbb{R}$-algebras which are not Banach algebras?

It is known that not every algebra (over a ground field $\mathbb{R}$ or $\mathbb{C}$) is a Banach algebra. It might be a silly question, but are there examples of finite dimensional ($\mathbb{R}$- or ...
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1answer
30 views

Is the result true when the valuation is trivial?

Here I proved the following result: Proposition: Let $(K,|\cdot|)$ be a valued field with non-trivial valuation and let $X$ be a vector space over $(K,|\cdot|)$. Two norms $p_1,p_2$ on $X$ are ...
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1answer
34 views

Distance Between Vectors

Generally, my main question is how to compute distance between two vectors. I'm aware of the formula $d=\| v-u\|$ where $v,u$ are two vectors, and $d$ donates the distance between them. More ...
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1answer
35 views

Prove that $(V,\lVert\cdot\rVert_\infty)$ is a Banach space

$X=C[0,1]$ (i) Prove that if the sequence $(f_n)_{n\ge1} \subseteq X$ converges to $f \in X$ in the supremum norm, then for each $t\in[0,1]$ one necessarily has $\lim_{n\to\infty} f_n(t) = f(t)$. ...
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11 views

regularized least squares (L1 norm)

My objective function that is to be minimized is as follows: $$\|y-Ax\|_2^2 + \alpha\|Lx\|_1$$ where $L$ is the gradient operator. Now this problem seems convex because the first term is quadratic ...
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1answer
34 views

Prove that N is a norm on $ℓ^3$

For each $z = (c_n)_{n≥1} ∈ℓ^3$, let $$N(z) =\bigg( \sum _{n=1}^{\infty} \frac{|c_n|^3}{|n|^3} \bigg)^{1/3}$$ Prove that N is a norm on $ℓ^3$. You may use without proof standard facts. Sequences ...
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43 views

Is $c_{00}$ closed in $(\ell^\infty,\|\cdot\|_∞)$

Consider the normed space $(X,\|\cdot\|)= (\ell^\infty,\|\cdot\|_\infty)$ and its linear subspace $V = c_{00}$ consisting of all sequences $(a_n)_{n≥1}$ of real numbers that eventually become zero: ...
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31 views

Derivative of squared norm of a complex function

Assuming $f(x)$ is a complex function, its squared norm is defined as $$|f(x)|^2 = f(x) . f^*(x)$$ What is derivate of $|f(x)| ^ 2$?
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8 views

Hamacher product in fuzzy logic

I was reading the article here Hamacher product is In page 41 in the article it describes three properties of Hamacher product. I do not understand the difference in notations and meaning of the ...
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1answer
15 views

Norm of rows of a matrix with a given spectral norm

Given a $n \times n$ matrix $M$ such that $\|M\|_2 = c$ (where $\|\|_2$ denotes spectral norm, or operator norm), is it true that for all $i = 1...n$ it holds that $$\sqrt{(\sum_{k=0}^n M(i,k)^2)} ...
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16 views

Norm of roots of unity in a subextension.

Suppose $L/k$ is a Galois extension of number fields, $G$ is the Galois group, and for $\frak p$ a prime ideal of $\cal O$$_L$, let $K=L^{G(\frak p)}$, where [$L$ : $K$] = $p$. Suppose $m$ is the ...
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1answer
19 views

Estimate for weak solution in elliptic equation

Let $B_1$ the unit ball closed in $\mathbb{R}^n$ $u: B_1 \to \mathbb{R}$ is a function in $H^1(B_1)$ If I have the following inequality: (*) $\|u\|_{L^{\gamma_i}(B_{\frac{1}{2}})} \leq ...
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1answer
25 views

Laplacian of powers of the norm in R^n

Consider the function $f(x) = ||x||^{2-n}$, $x \in \mathbb{R}^n$ \ $\{0\}$. I have derived the following expression for the Laplacian $$(2-n) n ||x||^{-n} - \frac{(2-n)n}{2} \sum^n_{j=1} ...
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31 views

Difference between $\ell ^{\infty}$ and $\ell^p$

With $p \in [1,\infty)$. With the $\ell^p$, the set is to do with summations but with the $\ell ^{\infty}$ it just says the supremum of a given vector right? Can someone explain why there is no sum in ...
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3answers
63 views

Prove $\int _a^b |f(t)| \, dt $ is a norm

Let $a < b$ be real numbers and $X = C[a,b]$ be the space of continuous functions $f : [a,b] → \mathbb R$. Prove that $$ \|f \|_1 =\int _a^b |f(t)|\,dt $$indeed defines a norm on $X$. Struggling on ...
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1answer
26 views

A criterion to determine if a matrix is invertible by its norm

I'm trying to prove an exercise, but I have no clue to start, any tip? This is the exercise: Suppose that a linear application $A:\mathbb{R}^n\rightarrow\mathbb R^n $ satisfy $\parallel ...
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0answers
18 views

Convergence criterion of vector fixed point iteration

As we know, the convergence criterion of scalar fixed point iteration, e.g. $x^{k+1}=f(x^k)$, is that $|f^{\prime}(x^*)|<1$. For the vector variable version, it has been proved in the case when ...
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23 views

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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23 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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1answer
49 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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1answer
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
34 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} || ...
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28 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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1answer
36 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
28 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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4answers
1k views

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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1answer
30 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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19 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
46 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 ...