Tagged Questions

1
vote
2answers
63 views

Proving that Euclidean space having the infinity metric is a complete metric space (stuck)

I am trying to prove that the space ${\mathbb{R}}^k$ with the $\infty$-metric is a complete metric space. I know that I need to show that every Cauchy sequence in the metric space ${\mathbb{R}}^k$ ...
7
votes
2answers
93 views

$C[0,1]$ is complete w.r.t. which norm(s)

$C[0,1]$ is complete w.r.t. which norm(s) $\displaystyle\|f\|_\infty=\sup_{t\in[0,1]}|f(t)|$ $\displaystyle\|f\|_1=\int_0^1|f(t)| \, dt$ $\displaystyle\|f\|_\infty^{0,1}=\|f\|_\infty+|f(0)|+|f(1)|$ ...
0
votes
1answer
44 views

Operator Norm of a Linear Transformation

PROBLEM For the linear transformation $T:\mathbb{R}^n\rightarrow\mathbb{R}^m$ equipped with the $l^1$-norm, namely, for $x\in\mathbb{R}^n$, $||x||=\sum_{j=1}^n |x_j|$ and similarly for ...
1
vote
3answers
65 views

Determining whether $f(x) = \frac{\sin||x||}{e^{||x||}-1}$ for $x \neq 0$, $f(x) = 1$ for $x = 0$ is continuous at $0$

$f: \mathbb R^m \to \mathbb R$ is defined as $$f(x) = \begin{cases}\dfrac{\sin||x||}{e^{||x||}-1} & \text{if $x \ne 0$} \\ 1 & \text{if $x = 0$.}\end{cases}$$ Note that $x$ is a vector in ...
0
votes
1answer
39 views

Prove that for every positive integer $d$ there exists $C(d)>0$ such that

for every polynomial $p(x)$ with degree $\leq d$, $\max\limits_{x\in[0,1]}|p'(x)| \leq C(d)\max\limits_{x\in [0,1]} |p(x)|$. There was also a hint given, that says to "use the compactness of a subset ...
-1
votes
1answer
72 views

Is $(\ell^1 , \| \cdot \| )$ a Norm space?

Suppose $ x= \{x_n \} \in \ell^ 1$ and $\| x \| = \sup | \sum_{k=1}^n x_k | $, let $ \|x\|_1 = \sum_{n=1}^{\infty} |x_n |$ is a norm for $ \ell^1 $ . Is $(\ell^ 1 , \| \cdot \| )$ a Normed ...
1
vote
1answer
43 views

$L_{k}^{1}([0,1])$ is a Banach space

Let $L_{k}^{1}([0,1])$ be the space of all $f\in C^{k-1}([0,1])$ such that $f^{(k-1)}$ is absolutely continuous on $[0,1]$ (and hence $f^{(k)}$ exists a.e. and is in $L^{1}([0,1])$). Show that ...
6
votes
3answers
137 views

If $\{x_n\}$ is a Cauchy sequence in a normed vector space, is $\frac{x_n}{\|x_n\|}$ Cauchy?

Let $\{x_n\}$ a Cauchy sequence in a normed vector space $X$. Is $$y_n = \frac{x_n}{\|x_n\|}$$ another Cauchy sequence in $D = \{x\in X : \|x\| = 1\}$? Remark: The idea is prove that if $D$ is ...
1
vote
1answer
89 views

Prove that the closed unit ball of $L^2[a,b]$ is closed in $L^1[a,b]$

Prove that the closed unit ball of $L^2[a,b]$ is closed in $L^1[a,b]$. My friends and I have literally been pouring over this problem for days now without success. We've been using Hölder's ...
1
vote
1answer
44 views

Does $\|z\|=(\sqrt{|x|}+\sqrt{|y|})^{\frac{1}{2}}$ define a norm on $\mathbb R^2$

Does $\| z\|=(\sqrt{|x|}+\sqrt{|y|})^{\frac{1}{2}}$, with $z=(x,y)\in\mathbb R^2$, define a norm on $\mathbb R^2$?
5
votes
1answer
188 views

Which of the following sets are open in $C^2[0,1]$? Explain. (Topology on normed spaces)

(a) $A= \{f \in C^2[0,1]:f(x)>0,\parallel f'\parallel_{\infty}<1, |f''(0)|>2\}$ (b) $B= \{f\in C^2[0,1]:f(1)<0,f'(1)=0,f''(1)>0 \}$ (c) $C= \{f\in C^2[0,1]:f(x)f'(x)>0$ for $0\le x ...
4
votes
4answers
119 views

Can a norm take infinite value? For example, $\|\cdot \|_1$?

A definition for norm from Wikipedia says Given a vector space $V$ over a subfield $F$ of the complex numbers, a norm on $V$ is a function $p: V → \mathbb{R}$ with the following properties: ...
1
vote
1answer
100 views

Simple function approximation of a function in $L^p$

I know that, in general, that any function $f \in L^p(X,\mathcal{M},\mu)$ can be approximated arbitrarily well by a simple function $\sum_{k=1}^n \lambda_k \chi_{E_k}$ where $a_k \in \mathbb{C}, E_k ...
1
vote
1answer
98 views

The vector space formed on C[0,1] and the norm $(\int_{0}^{1}|f(t)|^2 dt)^{1/2} $

How does one show that $(C[0,1], ||.||_{2})$ where $||.||_2=(\int_{0}^{1}|f(t)|^2dt)^{1/2}$ and $C[0,1]$ is the space of all continous function which are mapped from $[0,1]\rightarrow \mathbb{R}$, is ...
4
votes
2answers
127 views

Prove that $(B, \|-\|_{\infty})$ complete. B the set of bounded real valued functions on [0,1] which are pointwise limit of continuous functions.

Question: Prove that $(B, \|-\|_{\infty})$ is complete. B the set of bounded real valued functions on [0,1] which are pointwise limit of continuous functions on [0,1]. Context: Old exam problem I'm ...
2
votes
2answers
86 views

$\mathbb{R}^\infty$ is not complete in any $l^p$ norm.

As the title suggests, I would like to prove that the normed vector space $(\mathbb{R}^\infty , ||.||_{l^p})$ is not a Banach space, where $$\mathbb{R}^\infty :=\{ x:\mathbb{N} \rightarrow \mathbb{R} ...
2
votes
1answer
306 views

proving $L^\infty$ norm inequality (disprove $\Vert f\Vert_\infty\le\sqrt{n}$)

There are three parts in this question, I've done the first two but not sure about the third one. Also see $L^2$ norm inequality. In the third part, I am asked to show that if $W$ is a ...
3
votes
1answer
409 views

Relations between p norms

The p-norm is given by $||x||_{p} = (\sum_{n=1}^{\infty }|x_{n}|^{p})^{1/p}$. For $0 < p < q$, it can be shown that $||x||_{p} \geq ||x||_{q}$ (1, 2). It sppears that in $R^{n}$ a number of ...
-5
votes
2answers
307 views

Continuity in a normed space

Let $X$ be a normed space. Show that the function $f:X \to R$ defined by $f(x)=\|x\|$ is continuous on $X$.
2
votes
1answer
123 views

$C^1 [0,1]$ with different norm

If the space $C^1 [0,1]$ is equiped with norm $\Vert \cdot\Vert_1$,where $$ \Vert f\Vert_1=|f(0)|+\Vert f'\Vert _{C}=|f(0)|+\sup_{t\in [0,1]}|f'(t)| $$ for any $f\in C^1 [0,1]$, is this space Banach? ...
1
vote
1answer
78 views

Reproducing Kernel Hilbert Space- notation and basics

Am reading about Reproducing Kernel Hilbert Space(RKHS) while reading through Functional Analysis and Hilbert Space material and am unable to get the notation : $k(·,xi)$ correctly. What does the dot ...
2
votes
4answers
185 views

Product norm on infinite product space

Today I proved that if $V$ is a normed space with norm $\|\cdot\|$ then I can define a norm on $V \times V$ that induces the same topology as the product topology as follows: $\| (v,w) \|_{V \times V} ...
4
votes
1answer
244 views

Inequalities in $l_p$ norm

I'm having difficulty with the following problem. Any help would be appreciated. Problem: Consider the sequence spaces $l_p$ with the usual norm. If $1\le p\le q\le \infty$, I want to show the ...
12
votes
5answers
238 views

Passing from induction to $\infty$

Somehow, the operation of passing to the limit after I have shown that something is true by induction for each natural number $n$ troubles me each time. I know there are instances where one cannot ...
0
votes
0answers
45 views

existance of the interpolation space

Let $X\subset L_1+L_2$ and let $Y$ be interpolation space between $L_2$ and $L_{\infty}.$ Given $U:X\longrightarrow Y$. My question is the following: Is there exists space $Z\subset Y$, such that ...
4
votes
2answers
207 views

How convergence relates to equivalence of norms

Let $X$ be a normed linear space with two norms $||\cdot||_1$ and $||\cdot||_2$. Prove or disprove that this statements are equivalent: $||\cdot||_1$ and $||\cdot||_2$ are equivalent, $\{x_n\}$ ...
2
votes
1answer
53 views

P-adic “Norm” and scalability criterion

I just came across the p-adic norm for the first time. I tried to show that it is actually a norm on $Q$ but I was asking myself, whether checking scalability is a bit self referential ? What I mean ...
2
votes
1answer
115 views

Completing a normed space

Is the completion of $\{x=(x_n)|x_n\in \mathbb R \text{ and  for a given } x,\text{ only finitely many } x_n\neq0\}$ equipped with the norm $\|x\|:= |x_1|+|x_2|+...$ simply the set of all real ...
2
votes
1answer
134 views

The sup-norm of a diagonalizable operator

I want to get familiar with computing sup-norms of diagonalizable operators on $\mathbf{R}^n$. Suppose that I have a diagonalizable linear map $T:\mathbf{R}^n\to \mathbf{R}^n$ and I consider ...
6
votes
1answer
663 views

Does the p-norm converge to the max-norm in some norm

Does the $p$-norm on $\mathbf{R}^n$ converge to the max-norm on $\mathbf{R}^n$ as elements in the space of real valued continuous functions on $\mathbf{R}^n$ endowed with some norm? More precisely, ...
1
vote
1answer
69 views

How do you get pointwise convergence in the context of normed spaces [duplicate]

Possible Duplicate: Norm for pointwise convergence Let $V=C([0,1],\mathbf{R})$ be the vector space of continuous real-valued functions on $[0,1]$. Let $(f_n)$ be a sequence in $V$. Then ...
1
vote
1answer
274 views

Cauchy sequence in a normed space

Let $V$ be a real vector space. Suppose that $\Vert \cdot \Vert_1$ and $\Vert \cdot \Vert_2$ are two norms on $V$ which are equivalent. I suspect the following to be true. Let $(x_n)_{n=0}^\infty$ ...
4
votes
1answer
204 views

Non-completeness of the space of bounded linear operators

If $X$ and $Y$ are normed spaces I know that the space $B(X,Y)$ of bounded linear functions from $X$ to $Y$, is complete if $Y$ is complete. Is there an example of a pair of normed spaces $X,Y$ s.t. ...