0
votes
1answer
14 views

Does “uniformly isolated” imply closed?

Let $X$ denote a complete metric space and consider a subset $A \subseteq X$. Call $A$ uniformly isolated iff there exists $r > 0$ such that for all $a \in A$, we have that $B_r(a) \cap A = \{a\}$. ...
0
votes
1answer
66 views

Is it true that $0<0$ is a part of the proof for a unique fixed point of a contraction map?

I think it isn't, but my friend told me that this statement is true, he said that $0<0$ is even part of the proof for a unique fixed point of a contraction map. I google it but I couldn't find ...
2
votes
1answer
83 views

Completeness and separability of $B(X)$ (bounded linear operators on $X$)

Assume $X$ is a separable Banach space with norm $|| \cdot||$. Consider $\{f_n\}_{n \in N}$ a countable dense subset of $X$ and equip $B(X)$ (bounded linear operators on $X$) with the following ...
0
votes
1answer
87 views

Prove that $\exists\ C$ such that $T: C \to C$ with $C=\overline{\text{conv}}T(C)$

I have a problem: Let $K$ be a nonempty, closed, bounded and convex subset of reflective Banach space $X$. Suppose that $$T:K \to K$$ is nonexpansive. Prove that $\exists\ C$ such that ...
-1
votes
1answer
26 views

Finding a condition to make a map a contracting map

Consider $(\mathbb{C}^n ,||\cdot||_{1})$ where $||x||_{1} = \sum\limits_{i=1}^{n} |x_{i}|$ and $d_{1}(x,y) = ||x-y||_{1} = \sum\limits_{i=1}^n |x_{i} - y_{i}|$.
0
votes
0answers
17 views

Given data, approximations in a metric space for moving into a normed vector space isometrically.

Please see this question and this answer. Here $f_x(y)$ is approximated by $$x_v = [d(x,K_1),d(x,K_2),....d(x,K_N)]$$ by choosing to consider distances from $x$ to only certain points $K_i$ and ...
1
vote
2answers
96 views

Applying the Banach's Contraction Principle

I have a problem: For a system of linear equations: $$x_i=\sum_{j=1}^{n}a_{ij}x_j+b_i,\ \ i=1,2, \ldots , n \tag 1$$ Prove that, if $$\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}^2 \le q<1$$ then ...
1
vote
1answer
50 views

Finite range operator is compact

This theorem is from Rudin book which he says that obvious, but I'm quite confused how to prove it completely. Hope someone can help me clarify. Let $X$, $Y$ be Banach spaces, If $T \in ...
2
votes
1answer
23 views

Two ways to express boundedness

I'm little confused about the boundedness in a Banach space. Here are two boundedness definition we can encounter in Banach space: 1) A set $E$ is bounded if, for every neighborhood of 0, we have $E ...
3
votes
1answer
66 views

Is a Banach space $X$ Lipschitz equivalent to the metric quotient $X/B$, where $B$ is the closed unit ball?

Recall that the metric quotient $X/B$ is defined as follows: first we consider the equivalence relation $\sim$ on $X$ that identifies all points of $B$, then we define on the set of all equivalence ...
0
votes
2answers
74 views

Topology induced bycone metric

Is cone metric define atopology as same as the topology define by ametric? I have tried to prove it by theorems that joined them
2
votes
0answers
195 views

Proving the $l_p$ space is complete.

I'm trying to prove $l_p$ spaces are complete. We have an $l_p$ space $W$. Let us take a cauchy sequence. There exists $N_0\in\Bbb{N}$ such that for $m,n>N_0$, $d(x^m,x^n)<\epsilon$. This ...
3
votes
2answers
60 views

prove that $(\mathbb{R}^n,\|\cdot\|_\infty)$ is complete.

Suppose I have the metric spaces $(\mathbb{R}^2,\|\cdot\|_2)$ and $(\mathbb{R}^n,\|\cdot\|_\infty)$ where $\|x-y\|_2=\sqrt{\sum_{i=1}^2 (x_i-y_i) }$ and $\|x-y\|_\infty =\max ...
3
votes
1answer
82 views

geodesic metric

I'm trying to prove that the line segment is the minimizer of the distance $$d(x,y)=\inf l(\gamma),$$ where $x,y\in X$, $X$ is a Banach space, $\gamma$ is a path from $x$ to $y$ and ...
5
votes
3answers
192 views

Sequences are Cauchy depending on the norm?

Assume that we have two Banach spaces $B_1, B_2$ with their underlying sets being identical. Is it possible that a Cauchy sequence in one of the spaces would fail to be Cauchy in the other? I am ...
1
vote
0answers
70 views

Contraction Mappings

I'm self-learning functional analysis at the moment and although I can understand the underlying theory I have difficulty applying it in the aggregate. Can someone please break this down for me step ...
0
votes
1answer
55 views

A metric space $(\Bbb R,d)$ with $d(x,y)=||x-y||$ is complete!

I would like to receive only the hint, how to prove the statement on the heading. I understand that we have to prove that all Cauchy sequences converges in the space $\Bbb R$, e.g. ...
3
votes
1answer
319 views

If $X^\ast $ is separable $\Longrightarrow$ $S_{X^\ast}$ is also separable

Let $X$ be a Banach space such that $X$* (Dual space of $X$) is separable How can we prove that $S_{X^\ast}$ (Unit sphere of $X$*) is also separable Any hints would be appreciated.
0
votes
2answers
74 views

Need to confirm: Sup Metric $C[0,1]$, question about boundary

For the sup metric, $C[0,1]$. Let $S \subset C[0,1]$ be given by: $$S=\left\{f:[0,1]\to \mathbb{R} \ : \ 0 \leq f\left(\frac{1}{2}\right)<1\right\}$$ The question is simple: is this set open or ...
1
vote
1answer
87 views

Proving $\ell_\infty$ is complete

I start by taking a Cauchy sequence $(a_i)$ in $\ell_\infty$. I denote the terms of $(a_i)$ as $f_1, f_2, f_3, \dots$ and so on. For each $x \in \mathbb{N}$, the sequence $(f_1(x), f_2(x), f_3(x), ...
2
votes
0answers
135 views

Space of functions that are everywhere differentiable

Define the space $\beta^1([a,b])$ as the space of functions $f : [a,b] \mapsto \Bbb R$ which are everywhere differentiable and whose derivative $f'$ is a bounded function. One equips this space with ...
0
votes
1answer
129 views

Is the unit ball of a separable Banach space itself separable?

If $X$ is a separable Banach space, then do we know that its unit ball has a countably dense subset contained in the unit ball? This isn't obvious to me.
4
votes
1answer
577 views

Unit ball of a Separable Banach Spaces is metrizable

Please help me to understand why the unit ball of a separable banach space is metrizable, when it is given the induced topology from the weak topology. Specifically, my problem is I think I'd be able ...
3
votes
2answers
163 views

Completeness is property of the metric?

http://en.wikipedia.org/wiki/Banach_space From Wikipedia: In metric spaces, the completeness is a property of the metric. It is not a property of the topological space itself. If you move on to an ...
1
vote
1answer
172 views

Countable Hilbert Spaces

I have seen a simple proof that no banach space over $\mathbb{R}$ can be of countably infinite dimension. However since the space of all square integrable functions on the unit interval forms a ...
4
votes
2answers
70 views

Does $(x_n)$ Cauchy in $\ell^1$ implies $(\|x_n\|_1)$ is Cauchy in $\mathbb F$

Define $\ell^1=\{x\colon\mathbb N\to\mathbb F: \|x\|_1~\mbox{is finite}\}$ where $\mathbb F$ is either $\mathbb R$ or $\mathbb C$. If $(x_n)$ is a Cauchy sequence in $\ell^1$, does that mean that ...
3
votes
1answer
201 views

how to show that $c_0$ is complete

I want to show that the metric space $(c_0,d_\infty)$ is complete, where $c_0$ is the collection of all sequences $x\colon \mathbb N\to\mathbb R$ which tend to $0$. I have already shown that the space ...
2
votes
0answers
142 views

Convergence of a function in a metric space to its metric

Given a metric space $(\mathbb{A},d)$ with a metric $d$ being the Euclidean metric, if $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent sequence where $A$ is a matrix with the ...
2
votes
1answer
91 views

Balls and transformed sets in normed vector spaces

Let $T$ be a surjective, continuous linear operator between two Banach spaces $E$ and $F$. Assume that it is $B_F(y_0,4c)\subset \overline{T(B_E(0,1))}$, where $c>0$, $y_0 \in F$ ($B$ is for ...
1
vote
2answers
157 views

How to show $\alpha(A)\leq \beta(A)\leq 2\alpha(A)$

Let $X$ be a metric space and let $A\subset X$ be a bounded subset of $X$. I read on Wikipedia that the Hausdorff- and Kuratowski measures of non-compactness ($\alpha$, resp. $\beta$) satisfy the ...
2
votes
1answer
76 views

A question about weakening the conditions of Schauder's fixed point theorem

I'm currently doing a course on the theory of metric spaces. This is the version of Schauder fixed point theorem from my course: Let $(X,\|\cdot\|)$ be Banach and $C\subset X$ a closed, bounded, ...
1
vote
1answer
65 views

Something weaker than the Riesz basis

I have some function $f$, real valued and continuous. I formed functions $\{f_{m,k}, k \in \mathbb{Z}, m>0\}$ such that that $\mathrm{span}\{f_{m,k}, k \in \mathbb{Z}, m>0\}$ is dense in ...
2
votes
1answer
210 views

Completeness of $\ell^2$ space

I was reading up and it says that $(\ell^2,\|.\|_2)$ is complete. I know that a metric space $X$ in which every Cauchy sequence converges to an element of $X$ is called complete. And I know that a ...
0
votes
0answers
49 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 ...
2
votes
1answer
83 views

A question regarding convergence of distances to closed balls in Banach spaces

Let $X$ be Banach and let $B(x,\varepsilon)$ be the closed ball of radius $\varepsilon>0$ around $x\in X$ and consider the sequence $$f_{n;x}(y)= \begin{cases} 1-n\cdot d(yB(x,\varepsilon)), ...
4
votes
1answer
180 views

Measuring closed balls

Let $(X,\parallel \cdot \parallel)$ be Banach and $$\mathcal{BC}(X)=\{A\subset X\colon A \text{ is closed, bounded and non-empty}\}.$$ The natural metric on this space is the Hausdorff distance $d_H$ ...
1
vote
1answer
113 views

Convergence of a sequence of functions on closed balls

Let $X$ be a Banach space and $d$ be the induced metric. Let $S(x;r)$ denote the closed ball with radius $r$ at $x\in X$, that is,$$S(x;r)=\lbrace y\in X\colon d(x,y)\le r\rbrace.$$ Let $x,y\in X$ ...
1
vote
2answers
64 views

Is the set $E$ of sequences containing only entries $0$ and $1$ in $(m,\left \| \cdot \right \|_\infty)$ complete?

I can't really wrap my head around $E$, or a Cauchy sequence in $E$. I need to take a Cauchy sequence in $E$ and show it's Cauchy in $(m,\left \| \cdot \right \|_\infty)$? I think I can show $(m,\left ...
4
votes
1answer
1k views

Banach space of Lipschitz functions

Let $X$ be a compact metric space, and $F$ the space of all lipschitz functions $X \to \mathbf{C}$. Let $|f|_L$ be the least Lipschitz constant. We endow $F$ with the norm $||f|| = |f|_L + ...
1
vote
2answers
156 views

Sequences in $\ell_p$ spaces

Does there exist a sequence $(x_n)$ belonging to $\ell_1\cap\ell_2$ which converges in one but not the other? $(x_n)$ is of course a sequence in these spaces, so it's a sequence of sequences.