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

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Proving that function is norm [duplicate]

I have a problem with proving that $$ \|x\|_p=\left(\sum^{n}_{i=1}|x_{i}|^{p}\right)^{1/p} $$ is a norm where $p$ is number bigger than $1$ or $ 2$ the conditions are quite instant, but I can't ...
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Differentiating norm containing vectors and a matrix

I would like to differentiate $$D = ||L^{-1} (x-y)||_2^{2}$$, while x and y are vectors and L is a matrix Can someone show me how to do this? In other words, how to calculate: $$\frac{dD}{dx}$$ ...
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20 views

Stronger norm for continuous functionals

On $C[a,b]$ I want to show that $\|f\|_1 = \displaystyle \max_{a\leq t\leq b} |f(t)|$ is stroner than $\|f\|_2=\left(\int_a^b |f(t)|^2 dt\right)^{1/2}$ What I have done is as follows: $$ \begin{...
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Is norm $E[|X|^p]^{1/p}$ a continous function of $p$

Suppose $X$ is a random variable such that $E[|X|^p]<\infty$ is a function \begin{align} f(p)=E[|X|^p]^{1/p} \end{align} continuos function of $p$ for $1 \le p < \infty$. Here is my attempt to ...
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72 views

Alternative proof for dominated convergence theorem without using Fatou's lemma?

The conclusion of dominated convergence theorem is that $||f-f_n||_{L^1}\to 0$ as $n\to\infty$. After showing that $f_n\to f\in L^1$, why is it not possible to use the continuity of norm in order to ...
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Let $C\gt0$, I want to show that, $f(x)\leq C\|x\|,\quad \forall x\in X$

Let $X$ be a Banach space over $\Bbb R$ and let $f: X\to \Bbb R$ Where $f$ is sublinear and when $x_n\to x,\lim\displaystyle\inf_{n\to\infty} f(x_n)\geq f(x)$ Let $C\gt0$, I want to show that, $f(x)\...
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31 views

Does the norm of the matrix inverse alone say anything about the condition number

Is the fact that $$\| A^{-1} \|$$ is large, enough to conclude that the $cond(M)$ is large, where $cond()$ is the condition number of the matrix to determine if it is ill-conditioned. $$cond(M) = \| A^...
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57 views

Distance between two points wrt position and orientation

I'm looking for a distance function $d: \mathbb{R}^3 \times \mathbb{R^3}\rightarrow \mathbb{R}_0^+$ between two points given by $(x, y, \varphi)_{1,2}$, where $(x,y)$ is the position in two-...
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51 views

Norm equivalence and Banach spaces

If $(X,\|\cdot\|_1)$ is a banach space, and $\|\cdot\|_1$ is equivalent to $\|\cdot\|_2$, then $(X,\|\cdot\|_2)$ is a banach space. Does it also hold that if $(X,\|\cdot\|_1)$ and $(X,\|\cdot\|_2)$ ...
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What is the norm on the space $L^{\infty}([0,T];L^1(\Omega))$

The title is my question, so What is the norm on the space $L^{\infty}([0,T];L^1(\Omega))$
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39 views

Expressing continuity in terms of seminorms

Let $X$ and $Y$ be locally-convex topological vector spaces, with topologies given by families of seminorms $(p_i) _{i \in I}$ and $(q_j) _{j \in J}$, respectively. If $L : X \to Y$ is a continuous ...
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175 views

What is a norm topology in functional analysis?

I am currently reading up about norm topology, I have a background in functional analysis but I do not know anything about topology, aside from that topology is a collection of open sets with some ...
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42 views

Why does $1$-norm of a given vector form a diamond? [closed]

Why does $1$-norm of a given vector form a diamond? I came across this while watching a lecture on norms on youtube. Could not understand why this happens.
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Fixed point problem with matrix

I am looking for guidance for the following fixed point problem. $A$ is an $n\times n$ matrix. The rows of $A=(A_{i})_{i\in N}$ are in the set $A_{i}\in\mathcal{A}$, where $\mathcal{A}$ is the set ...
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50 views

sparse norm for optimization problem

I want to solve an optimization problem in general form: $$\arg \min f(x) + \lambda *g(x)$$ and i want to choose / define a $g(x)$ in a way to have a sparse solution such that between two possible ...
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25 views

How are absolute value of a field $F$ and norm of vector space $F/F$ related?

So I think it's true that a field $F$ and its respective vector space $F/F$ are isomorphic, since they consist of the same elements, and the operations of $F$ (addition,mult.) and $F/F$ (vector ...
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Showing two norms on $\mathbb{R}^n$ are dual

I am having trouble showing the following result. If $A$ is a positive definite matrix, then the norms (on $\mathbb{R}^n$) $\|x\|_A:= \sqrt{x^\top A x}$ and $\|y\|_{A^{-1}}:= \sqrt{y^\top A^{-1} y}...
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36 views

Linear functions $\mathbb{C}^n\longrightarrow\mathbb{C}^m$ are Lipschitz continous

Exercise: Show that any linear function from $\mathbb{C}^n$ to $\mathbb{C}^m$ is Lipschitz continous. (Hint: Use suitable norms.) I know that the maximum norm, the euclidean norm and the sum norm ...
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The $2$-norm of $A^*A$ is equal to the square of the norm of $A$

Show that $\|A^*A\|_2 = (\|A\|_2)^2$, where $A^*$ is the tranpose of $A$. My approach: $$\|A^*A\|_2 = \sqrt{\lambda_{\max} [(A^*A)^*(A^*A)]}= \lambda_{\max} (A^*A)=\|A\|_2^2$$ It that right?
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The $2$-norm of a submatrix does not exceed the norm of the matrix

Let $A \in \mathbb{R}^{n\times n}$ and let $1\le i_1 \le i_2 \le n$. Let $B$ be the submatrix with indices $(i_1:i_2,i_1:i_2)$. Prove that $\|B\|2\le \|A\|_2$. Can I say since $B$ is submatrix of $...
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109 views

Banach Lemma misunderstanding?

The Banach Lemma: Let $B$ be an n x n matrix . If in some induced matrix norm $ \|B\| < 1$, then $ I + B $ is invertible and $\|(I + B)\|^{-1} ≤ \frac1{(1− \| B \|)}.$ Question: Consider the ...
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Differential Equations: Jordan Form of a Matrix

I am using Lawrence Perko's book Differential Equations and Dynamical Systems, for my Differential Equations course. At the moment we are going over Jordan Forms of a linear system $x^{'}(t) = Ax$, ...
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A counterexample to $f_{n}\xrightarrow{\|\cdot\|_{L^{1}}}f$ implies $f_{n}\xrightarrow{\|\cdot\|_{C^{0}}}f$

$\|f\|_{L^{1}}\le\|f\|_{C^{0}}$ so $f_{n}\xrightarrow{\|\cdot\|_{C^{0}}}f$ implies $f_{n}\xrightarrow{\|\cdot\|_{L^{1}}}f$. I want to prove the converse is false but cannot come up with a ...
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26 views

Show that the operator $S\in B(Y)$ (the set of bounded linear operators from $Y\to Y$)

I am having trouble with the following problem: "Suppose we have an operator $S:Y\to Y$, where, $$S(g)(y)=g(y)-\int_0^1g(x)dx$$ and where we have $Y=C([0,1])$, equipped with the uniform norm ...
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36 views

Showing that all matrix norms satisfy the scalar property

How can I show that all matrix norms satisfy the following property (where $α$ is a scalar): $$\| α A \| = | α | \| A \|,\ ∀α ∈ R$$ This is for matrix norms defined in terms of $\| A \| = \max ( \|...
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75 views

Vector norm inequality [duplicate]

Is it true that for vectors $x$ and $y$ in $\mathbb R^n$ $|\Vert x\Vert -\Vert y\Vert| \ge \Vert x-y\Vert $? Can I simply use the triangle inequality $\Vert x\Vert +\Vert y\Vert \ge \Vert x+y\Vert $ ...
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Norm Equality for Map and Adjoint Map on Banach spaces

Case: for which I was studying first the case with $X,Y$ as Hilbert spaces but then later with $X,Y$ as Banach spaces. I am interested on the rigorous proof. I think the linearity and continuity ...
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Matrix Norm Proofs: Dropping the “max” term and denominator

To prove that $||A||_{\infty}≤\sqrt{n}||A||_{2}$, this math.exchange proof does the following: $$||A(x)||_{\infty}≤ ||A(x)||_{2}≤||A||_{2}||x||_{2}≤||A||_{2}\sqrt{n}||x||_{\infty}$$ Given the ...
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29 views

How to make the argument $\lim\limits_{k \to \infty} (a + b^k) = a, 0< b < 1$ for matrices

Suppose $a,b \in \mathbb{R}$, then we can easily see that $\lim\limits_{k \to \infty} (a + b^k) = a, 0< b < 1$ Suppose we have $A,B \in \mathbb{R}^{n \times n}$, $\|B\| < 1$ and attempt to ...
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If $\|A\| < 1$, does that imply $A$ is nilpotent?

Suppose $\|A\| < 1$ where $\| \cdot \|$ is the operator norm on matrices, intuitively, $\lim\limits_{k \to \infty} A^k$ goes to zero $\Rightarrow$ $A$ is nilpotent But is this indeed the case? ...
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Stuck on trivial proof show $(I - A)$ is non singular if $\|A\| < 1$

Assume $A$ is square, $\|A\| < 1$ Attempt: (by contradiction) Suppose $I - A$ is singular, then there exists $x$ such that $x \neq 0$, $\|x\| > 0$, $(I - A)x = 0$ Then $\|(I - A)x\| = 0$ and $...
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p - norms inequality

For $v \in \mathbb{R}^n$ denote by $||v||_p$ its p-norm. That is $(\sum_{i=1}^nv_i^p)^{\frac{1}{p}}$ where $v_i$ are the componenets of $v$. I'm looking for a way to bound the following expression: $$...
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38 views

Is there only one way to define a norm from an inner product?

Given an inner product $\langle,\rangle$, we can define a norm by $||x|| = \langle x,x \rangle^{\frac{1}{2}}$. My question is, are there other ways to derive a norm from an inner product space and if ...
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66 views

To prove norm on adjoint Banach space?

Case: which reminds me a lot about Bourbaki–Alaoglu theorem i.e. a unit case on a ball because of the last inequality and something with adjoint Hahn-Banach. My proposal by thinking the case on a ...
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164 views

Matrix p-Norm Inequality Proof

For $A\in \mathbb{R}^{m\times n}$, show: $\|A\|_\infty \le \sqrt{n} \|A\|_2$ I have seen the following proof on the forum: Say $e = [1,1,\dots,1]^T \in {\mathbb{R}}^{n} $, $\|A\|_\infty = \max_\...
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38 views

Show $\|f\|_{C[0,1]} \leq C \|f\|_{H^1[0,1]}$

My question: If $f \in C^1[0,1]$, show that $\|f\|_C[0,1] \leq C\|f\|_{H^1[0,1]}$, where $C$ is a constant independent of $f$ and $\|f\|^2_{H^1[0,1]}:= \|f\|^2_{L^2[0,1]}+\|f'\|^2_{L^2[0,1]}$. ...
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If $||A^N|| < 1$ then is ||A||<1?

following a previous question in functional analysis I asked Let $X$ be a banach space and $A$ is a bounded linear operator on $X$, and there exists some natural $N$ for which $ ||A^N|| < 1 $. ...
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If $||A^N|| < 1$, is $I-A$ invertible?

i have been given this question in functional analysis saying: Let $X$ be a banach space and $A$ is a bounded linear operator on $X$, and there exists some natural $N$ for which $ ||A^N|| < 1 $....
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Fenchel duality of infinity norm

The minimization problem is $\min\limits_{f_i} \sum^K_{i=1} \|f_i(\mathbf{p})\|_\infty$ Could someone explain how the Fenchel duality is used so the primal-dual formation becomes $$\min_{f_i} \...
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Progressed : Convergence problem in Hilbert Space and necessity of inner product

******** PROGRESS : so thanks to Ian's great comment I can get by the proof and that completeness is necessary but I need to know does this hold for general Banach spaces that are not Hilbert spaces? ...
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Is the max matrix norm induced?

Let $\|A \| = \max_{1 \le i,j \le n} |a_{ij}|$, where $A$ is a square matrix. I can prove that this is a matrix norm, but is it an induced norm?
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Norm inequality proof. [duplicate]

I need to prove the following inequality: $$\|x\|_{p} \geqslant \|x\|_{q}$$ if $q \geqslant p$. I tried to do it using this inequality: $$\frac{d\left(\sum_{i=1}^{d}|x|^{\alpha}\right)^{\frac{...
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Does the norm map respect diamonds of fields extensions?

Suppose we have fields $F_1, F_2$, and we form the compositum $E = F_1 F_2$. We then have a diamond of fields involving $E, F_1, F_2, F_1 \cap F_2$. Is it true that $$x \in F_2 \implies N_{E/F_1}(...
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47 views

Can someone explain why $\| \cdot \|_p, p = \frac{1}{2}$ isn't a norm?

$\| \cdot \|_p, p = \frac{1}{2}$ or "half norm" is not a norm What is a quick way to verify that it is indeed not a norm?
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19 views

Bound of norm of ordered numbers

Suppose we have two sets of real numbers $x$ and $y$ each with $n$ elements. Suppose I order the sets such that $x_{(1)} \geq x_{(2)} \geq x_{(3)} \geq ... \geq x_{(n)}$ and $y_{(1)} \geq y_{(2)} \...
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In which sense an expansive matrix is expansive

Let $A\in M_n(\mathbb{R})$. $A$ is called an expansive matrix if every eigenvalue of $A$ is strictly bigger than 1. From the name "expansive", one may deduce that (the space of square expansive ...
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80 views

Proving $\lVert x \rVert_3$ is smaller than euclidean norm

Is there a simple way to show that $\lVert x \rVert_3\leqslant \lVert x \rVert_2$ for vectors in $\mathbb{R}^2$? I've used the result in another problem but can't figure out where to get it from.
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An identity involving matrix norm and vector norm in relation to a density matrix in physics

This question is self-contained. For those who are interested, it arises from my study of the paper: K Lendi (1987), Evolution matrix in a coherence vector formulation for quantum Markovian master ...
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Visualising a region involving the $1$-norm

I'm a long-time lurker but first-time poster so I apologise if this post isn't formatted correctly. I'm having trouble visualising the region $\mathcal{R}$ defined as: $$\mathcal{R}=\{\textbf{x}\in\...
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91 views

How do you show that the norm for square matrices is submultiplicative?

In these notes: http://www.math.usm.edu/lambers/mat610/sum10/lecture2.pdf It says that square matrices satisfy so called submultiplicative norm $\|AB\| \leq \|A\|\|B\|$. Is it by definition or is ...