All Questions
Tagged with metric-spaces real-analysis
4,413
questions
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The existence of $f$ $k$-lipschitz in the subset $Y\subset \mathbb{R}$ implies the existence of a real $k$-lipschitz function $g$ such that $g|_Y=f$.
Here is the problem from a book for metric spaces I'm trying to solve:
Let $f:Y\rightarrow\mathbb{R}$ $k$-lipschitz in the subset $Y\subset \mathbb{R}$. Prove that there is a $k$-lipschitz function $...
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when do we say a metric space is quasi-invariant under a function?
A measure of a space that is equivalent to itself under "translations" of this space. More precisely: Let $(X,B)$
be a measurable space (that is, a set $X$
with a distinguished $ σ$
-algebra ...
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7
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203
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Problem about fixed points in a complete metric space
Let $(X,d)$ be a non-empty complete metric space and let $ f:X \rightarrow X$ be a function such that for each positive integer $n$ we have
(i) if $ d(x,y)<n+1$ then $d(f(x),f(y))<n$
(ii) if $d(...
1
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2
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71
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Real Analysis Question about Limit points and ε-neighborhoods
The question says "Prove that a point $x$ is a limit point of a set $A$ iff every ε-neighborhood of $x$ intersects $A$ at some point other than $x$."
I am having trouble proving the reverse ...
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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 ...
- 26.9k
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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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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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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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Can any open set in $\mathbb{R}^d$ be countably union of closed sets
I've already know that $\{B(x,r):x\in\mathbb{Q}^d,r\in \mathbb{Q}\} $ is an countable base of $\mathbb{R}^d$. Intuitively, I wonder that can an open set $\Omega\subset \mathbb{R}^d$ be countably union ...
2
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1
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Finishing the proof of the triangle inequality of Hausdorff metric
currently I am trying to show that a certain type of Hausdorff metric satisfies the following triangle equality and I am stuck.
Setup:
Take $(X,d)$ as metric space.
Denote by $C(X)$ the set of closed ...
0
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1
answer
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
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1
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Axiom of Choice in characterizing openness in subspace
Below is the typical characterization of open sets in a subspace $Y$ of a metric space $X$.
$E$ is $Y$-open iff there exists an $X$-open $S$ such that $E = S \cap Y$.
The forwards direction usually ...
- 119
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1
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36
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Reconciling metric and topological neighborhoods
Let $X$ be a metric space. Given a point in $x \in X$, an open neighborhood is more appropriately called an $\epsilon$-ball $N_\epsilon = \{p \in X : d(p, x) < \epsilon\}$, while a topological ...
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2
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3
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107
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Proving that the set of limit points of a set is closed directly [duplicate]
I'm working on Baby Rudin chapter 2's exercises and I'm stuck on problem #6, in particular the first part where he asks to prove that the set of limit points E', of a set E, is closed.
Here's my ...
- 199
1
vote
3
answers
90
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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 ...
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