# Tagged Questions

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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Two spaces which are isometric [closed]

prove that if X and Y are isometric and X is complete,then Y is also complete.Thanks
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### Is there any difference between Bounded and Totally bounded?

Is there any difference between bounded and totally bounded? (in a metric space)
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $\ell^{\infty}(\mathbb N)$ 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. ...
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### Can we have an isometric embedding of this metric space into an Hilbert space?

A metric space (from this Q&A), is defined below. I'd like to know if its possible to have an isometric embedding of this metric space into an hilbert space? As per Schoenberg theorem $-d^2(x,y)$ ...
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### addition and multiplication of functions in function space, continuous?

I have a norm that works in function space of C[0,1]. How do I show that addition and multiplication of functions (C[0,1]xC[0,1]->C[0,1]) are continuous functions?
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