real analysis - Why is $\ell^\infty (\mathbb {N})$ not separable ...
My functional analysis textbook says "The metric space l∞ l ∞ is not separable." The metric defined between two sequences {a1,a2,a3 …} {a 1, a 2, a 3} and {b1,b2,b3, …} {b 1, b 2, b 3,} is sup i∈N|ai …
functional analysis - $C (X)$ is separable when $X$ is compact ...
Jun 19, 2015 · this result is not trivial: If X is a compact T2 T 2 space X X, then C(X) C (X) is separable iff there is a metric X × X → R X × X → R that induces the topology of X X. You need to use the …
Prove that a subspace of a separable and metric space is itself separable
Prove that a subspace of a separable and metric space is itself separable Ask Question Asked 12 years, 3 months ago Modified 2 months ago
Definition of Separable Space - Mathematics Stack Exchange
Oct 8, 2020 · The standard definition (e.g. from wikipedia) that a separable topological space X X contains a countable, dense subset, or equivalently that there is a sequence (xn) (x n) of points in X …
$X^*$ is separable then $X$ is separable [Proof explanation]
Feb 5, 2020 · $X^*$ is separable then $X$ is separable Proof: Here is my favorite proof, which I think is simpler than both the one suggested by David C. Ullrich and the one I had ...
Is $L^p$ separable? - Mathematics Stack Exchange
Jun 27, 2014 · Wikipedia en.wikipedia.org/wiki/Separable_space#Non-separable_spaces: The Lebesgue spaces Lp, over a separable measure space, are separable for any 1 ≤ p < ∞.
galois theory - The definition of the separable closure of a field ...
Mar 7, 2024 · Non-separable extensions and elements are not so nice in some ways, in particular recall that an extension is Galois if it is normal and separable. So one might consider only considering all …
Prove if $X$ is a compact metric space, then $X$ is separable.
Prove if X X is a compact metric space, then X X is separable. Ask Question Asked 11 years, 3 months ago Modified 6 years, 9 months ago
Every subspace of a separable metric space is separable.
Dec 2, 2017 · IIf it were right it would apply to every separable space because you have not used any of the metric properties. But a separable non-metrizable space can have a non-separable subspace.
Proof of separability of - Mathematics Stack Exchange
A metrizable space is separable if and only if it is second-countable. This means that the euclidean topology on Rn R n has a countable basis. This countable basis is explicitely described in the first …