Questions tagged [metric-spaces]
Metric spaces are sets on which a metric is defined. A metric is a generalization of the concept of "distance" in the Euclidean sense. Metric spaces arise as a special case of the more general notion of a topological space. For questions about Riemannian metrics use the tag (riemannian-geometry) instead.
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Examples of function sequences in C[0,1] that are Cauchy but not convergent
To better train my intuition, what are some illustrative examples of function sequences in C[0,1] that are Cauchy but do not converge under the integral norm?
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Difference between limit point and limit
What is the difference between limit point of a sequence and limit of a sequence. Can it be unique?
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What is the difference between topological and metric spaces?
What is the difference between a topological and a metric space?
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How to prove that convergence is equivalent to pointwise convergence in $C[0,1]$ with the integral norm?
I'm trying to prove (or disprove) that in the set $C[0,1]$ of continuous (bounded) functions on the real interval [0,1] with the integral norm $\|f(x)\|_1 = \int_0^1|f(x)|dx$ that a sequence of ...
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Largest number of equidistant points
Given a metric space, we can check what the largest number of equidistant points are (ie, such that the distance between any two of these points is the same). Of course, this might not be finite, as ...
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The Class of Non-empty Compact Subsets of a Compact Metric Space is Compact
This is a question from my homework for a real analysis course. Please hint only.
Let $M$ be a compact metric space. Let $\mathbb{K}$ be the class of non-empty compact subsets of $M$. The $r$-...
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Proof of Triangle Inequality on $(\mathbb{R}^n, d_p)$
I have to prove the triangle inequality
$(|x_1 - z_1|^p + |x_2 - z_2|^p)^{1/p} \leq (|x_1 - y_1|^p + |x_2 - y_2|^p)^{1/p} + (|y_1 - z_1|^p + |y_2 - z_2|^p)^{1/p}$ for $p \geq 1$ on $\mathbb{R}^2$.
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$A,B \subset (X,d)$ and $A$ is open dense subset, $B$ is dense then is $A \cap B$ dense?
I am trying to solve this problem, and i think i did something, but finally i couldn't get the conclusion. The question is:
Let $(X,d)$ be a metric space and let $A,B \subset X$. If $A$ is an open ...
anonymous
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Why is it that $\mathbb{Q}$ cannot be homeomorphic to _any_ complete metric space?
Why is it that $\mathbb{Q}$ cannot be homeomorphic to any complete metric space?
Certainly $\mathbb{Q}$ is not a complete metric space. But completeness is not a topological invariant, so why is the ...
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Given a metric function between a set of abstract points, what is the best way to plot them on a 2D space?
I have a list of several entities, all of each have a numerical relationship to each other that defines an abstract distance.
Is there a mathematical way to plot all of these on a 2D space, turning ...
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Projection to space with smaller dimension that saves a distance
Suppose we have a countable set of objects $\{x_i|i \in [1..m]\}$ in a metric space $(\mathbb R^n,d_n)$ and a map ($F$) mapping the objects to objects in a metric space $(\mathbb R^1,d_1)$. For each ...
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A map on the unit ball
Let $B$ be the unit ball of $\ell^2(\mathbb{N})$, i.e. $B=\lbrace x\in \ell^2(\mathbb{N}): \|x\|\le 1\rbrace.$ For each $x=(x_1,x_2,\cdots)\in B$, let
$$f(x)=(1-\|x\|,x_1,x_2,\cdots).$$ Define $T:B\...
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Isometry in compact metric spaces
Why is the following true?
If $(X,d)$ is a compact metric space and $f: X \rightarrow X$ is non-expansive (i.e $d(f(x),f(y)) \leq d(x,y)$) and surjective then $f$ is an isometry.
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Unit sphere compactness in a metric space
Define $A$ as a nonempty set, $\mathcal{B}:=\{f: A \rightarrow \mathbb{R}: f(A) \text{is bounded} \} ,d_\infty:=\text{sup}\{|f(x)-g(x)|:x \in A\}$.
For which $A$ is $\overline {B_1(0)} \subset \...
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Closure of a subset in a metric space
Let $(X,d)$ be a metric space and $S \subset X$. Show that $d_S(x):=\text{inf}\{d(x,s): s \in S\}=0 \Leftrightarrow x \in \overline S .$
Notes: $\overline S$ is the closure of S. Maybe you can use ...
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