2
votes
1answer
34 views

Is the space $B([a,b])$ separable?

Let $a$, $b$ be two real numbers such that $a < b$, and let $B([a,b])$ denote the metric space consisting of all (real or complex-valued) functions $x=x(t)$, $y=y(t)$ that are bounded on the closed ...
1
vote
4answers
56 views

How to give a rigorous proof of this fact about closures of open balls in the euclidean spaces?

Let $n$ be a positive integer, $\vec{a} \in \mathbb{R}^n$, and $r > 0$. Then it is intuitively clear that the closuer of the open ball $$B(\vec{a} ; r) \colon= \{ \vec{x} \in \mathbb{R}^n \colon ...
1
vote
2answers
31 views

$C^n[a,b]$ as a normed algebra

I would like to prove that the space of complex valued functions, differentiable $n$ times with continuous derivative, $C^n[a,b]$, with the metric defined by the norm ...
4
votes
0answers
48 views

Compactness of a set of bounded functions in the uniform norm

Let $T$ be a non-degenerate compact interval in $\mathbb R$ and $f:\mathbb R^2\to\mathbb R$ a strictly concave function such that (a) $f(0,0)=0$, (b) $f$ strictly increases in the first argument, and ...
0
votes
1answer
9 views

Is there any metric $d$ of $\mathbb R^n$, $n<\infty$ such that $\mathbb R^n$ is bicompact and no norm induces $d$

There are some simple metrics can't yielded by norm .But add bicompact,I can't structure such example. In fact ,I want to know the condition of metric can be yielded by norm. Sorry for my poor ...
1
vote
2answers
27 views

$d$ is a metric space on $X\not=\{0\}$, obtained from a norm. $d'(x,y)=d(x,y)+1$. Show $d'$ cannot be obtained from a norm.

If $d$ is a metric on a vector space $X\not=\{0\}$ which is obtained from a norm, and $d'$ is defined by $d'(x,x)=0$, $d'(x,y)=d(x,y)+1, (x\not=y)$, show that $d'$ cannot be obtained from a norm. ...
1
vote
0answers
35 views

Regarding metrizability of weak/weak* topology and separability of Banach spaces.

Let $X$ be a Banach space, $X^*$ its dual, $\mathcal{B}$ the unit ball of $X$ and $\mathcal{B}^*$ the unit ball of $X^*$. The following result is well-known Theorem. If $X$ is separable, then ...
1
vote
0answers
20 views

Does a strictly convex and weak metrizable unit sphere of a Banach space imply separability?

I want know If $X$ be a Banach space with strictly convex norm and a metrizable unit sphere, $S_X=\{x\in X: \|x\|=1\}$, for the weak topology. Does a strictly convex and weak metrizable unit ...
2
votes
3answers
63 views

The “Circle” is a Vector Space?

Consider the set of angles $C = [0, \ 2\pi)$ and, for all $x,y \in C$, define the $sum$ operation as the sum modulo $[0, \ 2\pi)$. The identity element of the addition is the angle $0$. The inverse ...
1
vote
2answers
90 views

Continuity of Function Related to $F$-norms

Let $X$ be a locally bounded $F$-space and $\left\|\cdot\right\|$ be an $F$-norm on $X$. Suppose that $\left\|\cdot\right\|$ is concave: for all $x\in X$ fixed, the function ...
1
vote
0answers
36 views

Inequality proof using the triangle inequality

I am reading Kreyszig's Intro to Functional Analysis and am a bit stoked with one of the problems (problem 12 in section 1.1, page 9): Problem: Given a metric space $(X, d)$, show, using the ...
1
vote
1answer
33 views

Unit ball of continuous functions is a closed set - Proof with neighborhood argument

This question is trivial if one uses sequence definition, but I want to use the usual topological definition of closed set. That is , a set is closed if its complement is open. Let $U=\{f\in ...
1
vote
0answers
38 views

amazing boundedness problem from maximal function

Let $n\geq 2$. For any $M>1$, prove that there exists a constant $C_M>1$ such that for any ball $B$ in $\mathbb{R}^n$, if we denote $MB$ as the concentric ball of $B$ with $M$ times radius of ...
2
votes
1answer
51 views

Defining the integral on an arbitrary metric space

I am trying to prove a version of Mercer's Theorem for an arbitrary compact metric space; that is, I do not wish to restrict myself to the space of real-valued continuous functions $C[a,b]$. I ...
2
votes
0answers
60 views

$ d(x,y)^2 \le d(x,z)^2 - d(y,z)^2\color{}{+\varphi\big(x,y,z,d(x,y),d(y,z) \big)}? $

In a metric space $(M,d)$ the triangle inequality $d (x, z) \le d(x, y) + d (y, z)$ gives us's the inequalitie $$ \quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}} $$ ...
4
votes
1answer
54 views

Surjection of norms

Let $V$ be an infinite dimensional $\mathbb{C}$ (or $\mathbb{R}$) vector space. Suppose there exists two norms on $V$ such that \begin{equation*} \| \cdot\|_1 \leq \| \cdot \|_2. \end{equation*} Is ...
0
votes
2answers
34 views

triangle inequality to show metric

$d(x,y)= \begin{cases} 0 &\mbox{if } x=y \\ 1+\frac{1}{x+y} & \mbox{if } x\neq y \end{cases} $. Show that $(\mathbb{Z}^+,d)$ is a metric space. I'm stuck in proving triangle inequality.
2
votes
1answer
37 views

Difference between F-space and Frechet space in W. Rudin's “Functional Analysis”

In Walter Rudin's book, "Functional Analysis", we read that by talking about local base, he will be thinking about neighborhoods of $0$. In the vector space context, the term local base will ...
1
vote
1answer
109 views

Is sum of two metrics a metric?

The production of two metrics is a metric also. It's googled easy. But what's about a sum? As I can see sum is metric, as the triangle inequality of metric sum is the consequence of the inequality ...
0
votes
0answers
15 views

Sufficient conditions for RTree

What is the sufficient screening criteria of a space for the possibility to use R-Tree spatial index on it? I cannot apply it to a space with just Jaccard distance as the metric. As I suppose the ...
1
vote
0answers
63 views

Proving that there is no norm for the space of real-valued sequences making it a complete metric space.

Suppose I have a vector space $K$ which consists of real-valued sequences with only finitely many non-zero terms. I would like to show that there doesn't exist a norm on $K$ that would make it become ...
12
votes
1answer
189 views

Converse of a fixed-point theorem

I'm having some trouble furnishing a proof here. Let $(E, d)$ be a metric space such that any $k$-Lipschitz function has a fixed point for $0 < k < 1$. Does it follow, then, that $E$ is ...
4
votes
1answer
92 views

Trouble finishing a (direct) proof that $\ell^2(A)$ is a complete metric space

Let $A$ be any non-empty set. We can define summations of non-negative numbers over this index set by using a supremum of summations over finite subsets of $A$. That is, $$\sum\limits_{\alpha \in A} ...
2
votes
1answer
24 views

What is the metric on ${(L^{p}(\Omega))}^N$? (prove that sobolev spaces uniformly convex)

Here what I've done to prove that Sobolev Spaces $W^{(m,p)}(\Omega)$ are uniformly convex for $1<p<\infty$ Given integers $n\geq 1,k\geq0$, we define $N(n,m)$ as the number of multi-indices ...
2
votes
1answer
50 views

Is this space complete?

Let $X$ be the space of measurable functions $f:[0,1] \rightarrow \mathbb{R}$. I want to find out whether this space is complete under the metric $d(f,g):= \int_{[0,1]} \frac{|f-g|}{1 + |f-g|}$. Does ...
2
votes
1answer
111 views

Using the topology of uniform convergence for functions over non-compact spaces

Let $(X, d)$ be a (complete) metric space, and $C(X)$ be the space of continuous maps over $X$. If $X$ is compact, one often uses the topology of uniform convergence when analyzing $C(X)$. If $X$ is ...
0
votes
1answer
46 views

How to show that $(C^0((a,b)), d_\infty)$ is not a metric space

Let $d_\infty:C^0([a,b]) \times C^0([a,b]) \to [0,\infty)$ be defined as $$ d_\infty(f,g)=\sup\limits_{x \in [a,b]} \left\{ |f(x) - g(x)| \right\} $$ I have already shown that $(C^0([a,b]), ...
0
votes
1answer
36 views

Prove that metric space is complete

I have metric space: $$ X = <[0,+\infty), \rho>, \rho(x,y) = |ln(1+x) - ln(1+y)|$$ I know it is complete, but I don't know how to prove it. How can I prove that fact?
1
vote
1answer
27 views

Verify why this is not a metric $d(x,y)=\|x-y\|_p$ ( $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$).

$d(x,y)=\|x-y\|_p$ $p$ is prime and $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$, where $m, n$ are coprimes with $p$. This is not a metric because if $x=y=p^k\dfrac{m}{n}$, then $x-y=0=p^0\dfrac0n$. ...
3
votes
1answer
45 views

How to show the completeness of the space of Fourier transforms $\mathcal{F}L^{1}$?

Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ...
1
vote
0answers
39 views

a complete space

Define a set $$X=\left\{f:\mathbb{R}\rightarrow\mathbb{R}|f \mbox{ is n-times continuously differentiable}\right\}$$ equipped with the norm $$||y||=\max_{\begin{subarray}{l} ...
0
votes
4answers
137 views

Is there any difference between Bounded and Totally bounded?

Is there any difference between bounded and totally bounded? (in a metric space)
4
votes
3answers
103 views

generalization of Banach fixed-point theorem on short maps?

If $ \ T:X \longrightarrow X \ $ is contraction, then using Banach fixed-point theorem we know that the fixed point exists and all other points converge to that point. But what happens if $T$ is not ...
0
votes
0answers
49 views

Show the following extension is lipschitz

$X=S\cup\{x_0\}$, $f:S\rightarrow \mathbb R$ s.t. $|f(s)-f(t)|\leq kd(s,t)$ for $s,t\in X, k>0$. Suppose $x,y \in X$ s.t. $x\in S$ and $y\notin X$then $x=t, t\in S$ and $y=x_0$. I'm trying to ...
2
votes
2answers
64 views

What's wrong in this reasoning of $l_\infty$ separability?

While solving a problem, related to functional analysis, I've accidentally got a "proof" of $l_\infty$ being separable, tried to find a fault in it (as the result isn't true), but didn't succeed in ...
1
vote
1answer
61 views

Links between Minkowski metric, Hamming distance and Levenshtein distance

I know that Hamming distance is a particular case of Minkowski metric (with the specific definition of the subtraction). Also it seems that Hamming distance is a particular case of a Levenshtein ...
2
votes
1answer
91 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 ...
2
votes
1answer
175 views

$\ell^{\infty}(\mathbb N)$ is not a separable space

I have to prove that $\ell^{\infty}(\mathbb N)$ is not separable. My attempt Consider a SUBSET $V$ of $\ell^{\infty}(\mathbb N)$ consisting of bounded sequences that have only $0$, $1$ entries, e.g. ...
2
votes
1answer
65 views

Separability in Metric Spaces

Let $(X,d)$ be a non-separable metric space. My question is the following: does there exist some $\epsilon > 0$ and some uncountable subset $S$ of $X$ such that $d(x,y) > \epsilon $ for any $x, ...
3
votes
1answer
132 views

Show this metric generates the product topology on $X$

Let $(X_n, d_n)$ be a sequence of metric spaces. Show that the function $ d: X \times X \to \mathbb R^+$ on the product space $X: = \prod_n X_n$ defined by $$d ((x_n)_{n = 1}^\infty, ...
0
votes
1answer
32 views

How to show in general a certain type of metric generates a certain type of topology?

If we're given a set $X$ together with a metric $d: X \times X \to \mathbb R^+$, and a topology $ \mathcal F \subset 2^X$. What do we need to do in order to show this metric generates this topology?
0
votes
1answer
53 views

Show that $\ c_X(p,q) \le d_X(p,q)$, for $ p, q \in X$

Update I'm trying to show the Corollary, but I have stuck...That is: For any complex space $X$, we have: $$\begin{align} (1).\ c_X(p,q) &\le d_X(p,q),\ \text{for}\ p, q \in X \\ (2).\ ...
0
votes
0answers
55 views

Sequence of $r_n:=\frac{x_{n+1}}{x_n}$ where $x_n$ is a sequence

Let $f:X\rightarrow X$ be a function on some complete metric space $(X,d)$. Let, $x_n$ is a sequence in the metric space defined by $x_{n+1}=f(x_n)$ and starting from $x_0$. My questions are (1) ...
0
votes
1answer
89 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
vote
1answer
86 views

When is $\mathcal C(X)$ complete?

I learned today in class that $\mathcal C([a,b])$ is complete with the supremum norm. That is, any uniformly convergent sequence $(f_n)$ is Cauchy, and the converse is true. I asked my teacher about ...
2
votes
4answers
195 views

Concerning $C^0[0,1]$ and the $L^1$-Norm.

Consider the well known Euler sequence of functions $x^n$ ($n=1,2,3\ldots$) on $[0,1]$. It is clear that it converges against $\chi_{1}$, the characteristic function of the singleton $1$, in the ...
2
votes
2answers
64 views

Topology induced by Scalar Product

I just had an idea: It is clear that every scalar product induces a norm and that a metric and that finally a topology. Turning this argument around we know: Not every topology induces a metric only ...
2
votes
1answer
440 views

Why is $l^\infty$ not separable?

My functional analysis textbook says "The metric space $l^\infty$ is not separable." The metric defined between two sequences $\{a_1,a_2,a_3\dots\}$ and $\{b_1,b_2,b_3,\dots\}$ is ...
1
vote
1answer
30 views

Distance between Unilateral shift and invertible operators.

I want to prove that the distance between unilateral shift and normal operators is $1$. But I need to prove that $d(S,\operatorname{Inv}(L(H))= 1$, where $H$ is a Hilbert space. Does anyone have any ...
0
votes
1answer
116 views

Convergence in $L^\infty$ norm and continuous function

Let $\mathcal{C}(T)$ be the set of continuous functions on $T$, which is a metric space under the norm $\left\|f\right\|_{\infty}=\sup_{t\in T}\left|f(t)\right|$. Suppose $\{X_{n}\}$ and $X$ take ...