Tagged Questions

A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.

41 views

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 ...
17 views

Prove that the limit of a specific sum of sequences in a normed space is 0

Assume that m sequences $(u_n^k)_{n=1}^\infty$ ($k=1, \ldots, m$) in a normed space satisfy $$\lim_{n\rightarrow\infty} \left(\|\sum _{k=1}^m u_n^k\| - \sum_{k=1}^m \|u_n^k\|\right) = 0$$ Show that ...
46 views

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: ...
34 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 ...
51 views

44 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?
44 views

Why in normed vector spaces we can define infinite series but in metric space we can not?

We usually define infinite series by partial sums and an inifinite series is said to converge if its partial sum converges. So, if $X$ is a normed vector spaces and $s_n=x_1+...+x_m$ is a partial sum ...
91 views

What is a “pseudonorm”?

The following is an excerpt of a note in topological vector spaces. I have tried to search "semi-pseudonorm" on Google but I have got nothing so far. A search with "pseudonorm" returns what we ...
14 views

59 views

Why is the norm ball a square in $\,\mathbb R^2\,$ under $\,l^\infty\,$ norm?

Suppose $\,x = \left(x_1, x_2\right)$, then $\,l^2\,$ norm ball is $\,\left\lbrace x\;\big\vert\;\, \sqrt{\left\lvert x_1 \right\rvert^2 + \left\lvert x_2 \right\rvert^2} \leq 1\right\rbrace$ Easily ...
40 views

Is the norm ball a set or the boundary of a set?

Recall normed ball in $R^2$ under different norms is typically intuited as follows But looking at someone of the definition of normed ball it seems that it describes a closed set rather than the ...
17 views

$inf\{||x|| : x \in X , ||F(x)|| = \alpha\} = \frac{\alpha}{||F||}$

If $F \in B(X,Y), F\neq 0$ and $\alpha \geq 0$, then show that $$inf\{||x|| : x \in X , ||F(x)|| = \alpha\} = \frac{\alpha}{||F||}$$ where $B(X,Y)$ is the set of all bounded functions from $X \to Y$ ...
44 views

Convolution inequality, Gaussian convolution

I'm reading a proof that says for $f\in L^p$ with $p\in[1,\infty)$ we have $\|f\ast p_t-f\|_p\to 0$ as $t\to 0$, where for $t>0$, $p_t(x)=\frac{1}{(2t\pi)^{\frac{d}{2}}} e^{-\frac{\|x\|^2}{2t}}$ is ...
24 views

Norm of homomorphism embedding Banach algebra into matrix algebra

Let $A$ be a nonunital Banach algebra, and let $\tilde{A}$ be the unitization of $A$, i.e. $\tilde{A}=\{(a,\lambda):a\in A,\lambda\in\mathbb{C}\}$ with multiplication given by ...
66 views

Arzelà–Ascoli theorem for operators

If you have a net of continuous linear operators between reasonable spaces (complete, at least), does there exist Arzelà–Ascoli-like theorems giving convergent subnets? I believe that it should be ...
78 views

Difficulty understanding the proof of equivalence of all norms over $\mathbb R^{n}$

To prove that all norms are equivalent on $\mathbb R^{n}$ , the book I am reading , first takes an arbitrary norm $$|\ \ | \ :\ \mathbb R^{n}\rightarrow\ \mathbb R$$ and then ...
88 views

In the proof that $L^{1}$ norm and $L^{2}$ norm are equivalent.

To prove that $L^{1}$ norm , denoted by $||\ \ ||_{1}$ and $L^{2}$ norm , denoted by $||\ \ ||_{2}$ are equivalent we have to find constants $C_{1},\ \ C_{2}$ that satisfies ...
34 views

Prove that $\| x \| \le \alpha {\| x \|_{sum}}\,,\,$ $\forall x\in R^n\;$ such that $\alpha=\max_{1\le i\le m}\{\|e_i\|\}\,,\;$ $\|\cdot\|=$ any norm.

Prove that $$\| x \| \le \alpha {\| x \|_{sum}}\quad,\quad \forall x\in \mathbb{R}^n$$ for any norm $\| {\, \cdot \,}\|\quad$ , $\quad{\left\| x \right\|_{sum}}$ is the norm of the sum $\quad$, ...
36 views

Strategy for establishing the triangle inequality of a seminorm

One proof that the $p$-norm $\| x\|_p = (|x_1|^p + \ldots + |x_n|^p)^\frac{1}{p}$ satisfies the triangle inequality exploits the fact that $x \mapsto |x_1|^p + \ldots + |x_n|^p$ is a convex ...
33 views

How to show continuity of $\cdot$ in a normed vector space?

Let $(V,+,\cdot)$ be a normed vector space with the underlying field $K$ . We have to show that $+,\cdot$ are continuous functions. Since $V$ becomes a metric space under this norm so I can use ...
47 views

Equivalent norms on $C[0,1]$

For each $f\in C[0,1]$ set $$\|f\|_1 = \left(\int_0^1 |f(x)|^2 dx\right)^{1/2},\quad\quad \|f\|_2 = \left(\int_0^1 (1+x)|f(x)|^2 dx\right)^{1/2}$$ Then prove that $\|\cdot\|_1$ and $\|\cdot\|_2$ are ...
32 views

76 views

How to show that every Cauchy sequence converges in some normed linear space?

(Hunter and Nachtergaele, 1.6) Using the fact that R is a complete metric space with respect to Euclidean distance, show that Rn is a Banach space when equipped with (a) the Euclidean norm (b) the ...
36 views

Completeness of metric, normed and inner product spaces

For a metric space to be complete, it needs to have all cauchy sequences converge in the metric. 1) For a normed space to be complete, does it need Cauchy sequences to converge in the norm or in the ...
47 views

Is the spectral radius of a matrix a convex norm of it?

I am wondering if the spectral radius of a matrix is may be some kind of a norm ($l_{\infty}$-norm?) of it and if that is convex. Any pointers to related ideas would be helpful too.
77 views

If a linear operator between two normed linear spaces is continuous at one point, then it is continuous at all points.

Let $f : \langle V_1, \|\cdot\|_1\rangle \to \langle V_2, \|\cdot\|_2\rangle$ be linear. Then if $f$ is continuous at some $v \in V_1$, then it is continuous on all of $V_1$. Without appealing to ...