Questions tagged [separable-spaces]
For questions about separable spaces, i.e., topological spaces containing a countable dense set.
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Theorem 3.6.4 (Separable space) Kreyszig.
Kreyszig , in "Introductory Functional Analysis with Applications" , has this Theorem 3.6.4 (with Proof) concerning Separable spaces.
Theorem. Let $H$ be a Hilbert space. If $H$ contains an ...
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Two definitions of separability?
I am confused if the following two definitions have anything to do with each other:
In analysis IV we have seen that a space $X$ is called separable if it has a subset which is countable and dense, i....
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Dense Sequences
I'm trying to generalize the following result:
There exists a sequence whose set of subsequential limits is equal to $\mathbb{R}$.
Using the following lemma,
Given a sequence $(p_n)$ with range $E$,...
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$C^n(X)$ separability
Using Stone-Weiestrass thm we can say easily that $(C^n(K), \| . \|_{\infty})$ where $K \subset \mathbb{R}^n$ compact, is separable as subspace of separable metric space.
Anyway i'm interested to the ...
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When is $f = X^4 -1 \in \mathbb{F}_p[X], p $prime, irreducible and/or seperable? [duplicate]
I'm having some trouble figuring out a solution to this. I understand that $f$ is separable, iff all its roots are distinct, however I'm completely clueless about how to investigate that criterion......
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Can the separability of a space depend on the axiom of choice?
Does there exist a topological space $X$ such that in $\mathsf{ZFC}$, $X$ is separable, but such that it is consistent with $\mathsf{ZF}$ that $X$ is not separable? The motivation behind this question ...
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Seperability for the collections of all non-empty compact subsets of $\mathbb{R}^2$ with Hasudorff metric
Let $X$ be the collections of all non-empty compact subsets of $\mathbb{R}^2$, which has the Euclidean metric. Let $(X,d)$ be a metric space, where $d$ is the Hasudorff metric.
Is $X$ separable?
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Proving a Proposition about separable Hausdorff spaces that are locally euclidean
Let X be a separable Hausdorff space such that for every $x \in X$ there exists an open neighborhood $U$ of $x$ such that $U$ is homeomorphic to an open subset of $\mathbb{R}^n$.
Show that:
(i) $X$ is ...
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If $X$ is a compact separable topological space with a countable family of complex valued continuous functions, then $X$ is metrizable
I am studying measure theory, topology, and functional analysis in mathematics. Let $X$ be a compact topological space. We assume that there exists a countable family $\{f_n\: X \to \mathbb C: n\in\...
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Show that Banach space valued function is measurable.
I am given a seperable Banach space and an interval $J$. Pettis' theorem tells that for such a space $Y$, we have: $f : J \rightarrow Y$ is measurable iff for any bounded linear functional $g \in Y^*$,...
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Strong Seperation between hypreplane and an affine subspace
Let $h \in \mathbb{R}^n\backslash \{0\}$ and $r \in \mathbb{R}$. A sure that $M$ is an affine subspace of $\mathbb{R}^n$ with $H(h, r) \cap M=\phi$. I tried to use Banach separation theorem to show ...
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There exists the converse of this corollary from Brezis?
In Brezis's Functional Analysis, there is a corollary
Corollary 3.30. Let $E$ be a separable Banach space and let $\left(f_{n}\right)$ be a bounded sequence in $E^*$. Then there exists a subsequence $...
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A separable normed space that is continuously embedded in a non-separable normed space implies that this embedding isn't dense.
As a preliminary I introduce the definition of denseness I am using:
Definition (dense subsets of metric spaces). Suppose $(M,d)$ is a metric space. A subset $S \subset M$ is called dense in $M$ if ...
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Separability of codomains of Borel functions taking values in completely regular spaces
I am looking for a reference (or a counterexample) to the following statement.
Let $X$ be a separable metric space. Suppose that $Y$ is a completely regular topological space and $f\colon X\to Y$ is a ...
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Adapting a proof of the non-separability of Morrey Spaces for a different definition.
In the article "Morrey spaces, their duals and preduals", by Marcel Rosenthal and Hans Triebel, for every $1 \leqslant p < \infty$ and $-\frac{n}{p} < r < 0$ the Morrey Spaces are ...
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