All Questions
Tagged with metric-spaces analysis
1,236
questions
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56
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If $\forall n,\sum_ka_{n,k}^2<\infty$ and $\forall k,a_{n,k}\to b_k$, how to show that $\sum_kb_k^2<\infty$? [closed]
Let $\ell^2$ denote the metric space of all the square-summable sequences of real numbers. Let $p_n = \left( a_{n1}, a_{n2}, a_{n3}, \ldots \right)$ for $n = 1, 2, 3, \ldots$ be a sequence of points ...
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34
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There is at least one point of every non-empty open subset of the $\ell^2$ space whose first coordinate is nonzero [duplicate]
Here we take
$$
\mathbb{N} := \{ 1, 2, 3, \ldots \}.
$$
Let $\ell^2$ denote the set of all the real (or complex) sequences $\left( \xi_i \right)_{i \in \mathbb{N} }$ such that the series $\sum \left\...
- 26.9k
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69
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The function $d \colon \mathbb{R}^2 \longrightarrow \mathbb{R}$ defined by $d\big((a,b)\big)=\lvert x-y\rvert$ is continuous [duplicate]
Let the function $d \colon \mathbb{R}^2 \longrightarrow \mathbb{R}$ be defined by
$$
d\big( (x, y) \big) := \lvert x-y \rvert \qquad \mbox{ for all } (x, y) \in \mathbb{R}^2.
$$
Let $\mathbb{R}$ and $...
- 26.9k
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41
views
The diameter of the union of two sets in a metric space cannot exceed the sum of the diameters of the two sets and the distance between them
Let $A$ and $B$ be any two (nonempty) sets in a metric space $(X, d)$. Then how to show that
$$
d (A \cup B) \leq d(A) + d(B) + d(A, B)? \tag{0}
$$
Here we have the following definitions:
For any (...
- 26.9k
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53
views
Given a metric $d$, is continuity of $f$ absolutely essential for the composite function $f \circ d$ to be a metric (equivalent to $d$)? [closed]
Let $d$ be a metric on a nonempty set $X$, and let $f \colon [0, +\infty) \longrightarrow [0, +\infty)$ be a function such that (i) $f(0) = 0$, (ii) $f(r+s)\leq f(r) + f(s)$ for all $r, s \in [0, +\...
- 26.9k
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1
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35
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Which metrics (on vector spaces) can be induced?
Is there a way to classify which metrics defined on vector spaces can be induced by a norm? ie. there exists norm $n: X\to \mathbb{R}$ on the vector space $X$ such that the metric $d(x,y)=n(x-y)$.
I ...
- 477
-1
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24
views
Distances on Cartesian product [duplicate]
I was studying general topology when a question came to my mind.
Assume given n metric spaces, call $X$ their Cartesian product and define three real-valued functions from $X\times X$:
the first ...
- 579
-2
votes
1
answer
101
views
Convergence of $\sum^{\infty }_{n=0} |a_{n}-b_{n}|$ from convergence of $\sum^{\infty }_{n=0} |a_{n}|$ and $\sum^{\infty }_{n=0} |b_{n}|$
Exercise 12.1.15 of Tao's Analysis II book asks the reader to show that the function $d:X\times X\rightarrow \mathbb{R} \cup \left\{ \infty \right\} $ defined by $d\left( a_{n},b_{n}\right) =\sum^{\...
2
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answers
33
views
Linear Analysis – Examples 1-Q5 General $l^p$ spaces vs $l^1, l^\infty$
Let $1<p<\infty$, and let $x$ and $y$ be vectors in $l_p$ with $\left \|x \right \|=\left \|y \right \|=1$ and $\left \|x +y \right \|=2$, how to prove $x=y$?
I know how to prove for $p=2$ ...
- 477
1
vote
3
answers
90
views
Proving that the closure of a set is closed directly
Currently working through Rudin's principle's of mathematical analysis. I am trying to prove directly that the closure of a set is closed but am hitting a wall on one part of the proof. Namely, if we ...
- 199
3
votes
3
answers
79
views
Understanding the Proof of Relative Openness Theorem in Rudin's Principles of Mathematical Analysis
I'm having trouble understanding Rudin's proof for the theorem stating:
"Suppose $Y \subseteq X$. A subset $E$ of $Y$ is open relative to $Y$ if and only if $E = Y \cap G$ for some open subset $G$...
- 199
7
votes
1
answer
121
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Upper semicontinuity of sequence of Hausdorff measures
This is exercise 12.2 of Measure theory and integration of Michael Taylor.
Let $(d_j)$ be a sequence of metrics on a compact space $X$ such that there exists $c,C>0$ with
$$ cd_0(x,y) \leq d_j(x,y) ...
- 636
1
vote
1
answer
91
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Prove that if $K \subseteq X$ is compact and $F \subseteq X$ is closed, then $K \cap F$ is compact
Let $(X,d)$ a metric space. Prove that if $K \subseteq X$ is compact and $F \subseteq X$ is closed, then $K \cap F$ is compact
Here's my proof so far, please check it.
Proof.
Assume that $K \subseteq ...
- 19
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answers
36
views
May I say that if $(M,d)$ is a metric space and $K \subseteq M$ is closed, then $K$ dist(a) is a minimum (and vice versa)?
I am slightly stuck on the proof of the following proposition: is it ok to look at the cases $K$ is open and $K$ is closed seperately, and if so, does it work in my favour as I claim?
Let $(M,d)$ be ...
- 21
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33
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continuity of a function defined in a compact metric space
Let $K$ be a compact metric space and $S\subset K$ a compact subspace such that
$$S \subset \bigcup_{i=1}^{k} B_{s_i}$$
where each $s_i \in S$ and $B_{s_i}$ are open balls centered at $s_i$
consider ...
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