# 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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### 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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### 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 (...
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### 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 $p=2$ ...
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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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