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

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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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37 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
45 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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1answer
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
33 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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30 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: $$ ||V|...
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1answer
46 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 f,...
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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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2answers
102 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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59 views

Is the matrix norm of a matrix equal to the maximum of the norms of its Jordan block?

Let $J$ be a Jordan block matrix with blocks $J_1,\cdots,J_n$. I came up with some examples of $J$ and noticed that $\|J\|=\max_{i=1,\cdots,n}\|J_i\|$. Does this result always hold? The norm I use ...
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1answer
34 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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2answers
43 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
32 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
36 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
36 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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0answers
12 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
36 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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2answers
46 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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1answer
66 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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11 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
20 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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19 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
21 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 C\|u\|_{L^2(...
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1answer
27 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} x_j^{-\frac{...
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0answers
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
65 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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30 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 A-Id\...
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24 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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27 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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33 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
51 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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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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33 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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39 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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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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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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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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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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34 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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33 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 ...
3
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2answers
47 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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22 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 $$\frac{\|(A+M)^{-1} - A^{-1} \|}{\|A^{-1}\|}\le 2c.$$...