All Questions
Tagged with metric-spaces complete-spaces
500
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Completeness meaning (complete basis vs complete metric space)
Today my professor started talking about the formalism of QM.
We talked about the eigenvectors of a Hermitian operator (over Hilbert space) as a "complete set". He also mentioned briefly ...
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2
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1
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Infinite-dimensional cube $[0,1]^\mathbb N$ compact/complete under certain metrics?
Consider the infinite-dimensional cube $[0,1]^\mathbb N := \{ (x_i)_{i=1}^\infty \ | \ 0 \le x_i \le 1 \ \forall i \}.$ This space can be equipped with certain metrics. Below are two of them.
$d_\...
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Always Closed Metric Space is Complete
I am trying to prove that
Given a metric space $Y$, if for every metric space $X \supseteq Y$, $Y$ is $X$-closed, then $Y$ is complete.
by contradiction or contraposition, such that I don't use the ...
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A complete metric space contains a convergent sequence or an infinite discrete subset
Theorem. Let $X$ be an infinite complete metric space. Then there is an injective convergent sequence in $X$ or there exists $\varepsilon>0$ and an infinite $\varepsilon$-discrete subset $A\...
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What happen if we eliminate only one point from a Cauchy complete metric space?
Consider $\mathbb{R}$ with the usual metric. We know that $\mathbb{Z}$ is complete as a subspace of $\mathbb{R}$ (because is closed), and $\mathbb{Z} \setminus \{0\}$ is still complete. But if we take ...
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Does complete and separable Wasserstein space imply the completeness of the base space?
Also asked on MathOverflow.
Let $(Z,d)$ be a metric space, and for $p\geq 1$, consider a metric space $(W_p,d_{W^p})$ defined by
The Wasserstein Space $\begin{align}W_p = \{\mu|\mu\textrm{ is a Borel ...
- 1,565
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Why metric equivalence does not preserve completeness but norm equivalence does? [duplicate]
I was studying Functional analysis when I came across the normed space equivalence i.e. equivalent norms. I already proved that equivalent norms on a vector space $X$ over field $\mathbb{R}$ or $\...
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Completeness preserved under specific homeomorphism
Problem: Let $(X,d)$ be a complete metric space, $(Y,d')$ a metric space, and $f\colon X\to Y$ a homeomorphism, such that there exists a $c>0$, for which $$c\cdot d(x,y)\leq d'(f(x), f(y)).$$ Show ...
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Find an example to show Cantor intersection theorem does not hold if we eliminate one condition
Cantor's intersection theorem states that:
Let $(X,d)$ be a complete metric space, and let $A_1\supset A_2\supset...$ be an infinite chain of nonempty, closed, bounded subsets of $X$. Suppose further ...
user1277306
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Subspace of real valued functions on [0,1] is complete
I’m currently working on the following problem:
Let $X$ denote the set of nondecreasing functions $f:[0,1]\to\mathbb{R}$. We endow $X$ with the sup metric. Prove that $X$ is complete.
I notice that if ...
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Can we include a metric space into its completion?
Let $(X,d)$ be a metric space and let $(X^*,d^*)$ denote its completion (via equivalence classes of Cauchy sequences in $X$). Let $f:X \to X^*$ be an isometry such that $f(X)$ is dense in $X^*$.
Now ...
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Compact-open topology not always complete with Arens' metric
For an infinite-type surface $S$ with complete metric $d$ and compact exhaustion $\{K_n\}_{n=1}^\infty$, Vlamis, in "Notes on the Topology of Mapping Class Groups" Appendix A, defines a ...
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Using only the universal property, prove that $X$ is dense in its Cauchy completion
Long ago, I learned about the Cauchy completion of metric spaces via the usual explicit construction of quotienting the set of Cauchy sequences. For a metric space $X$, let $\hat X$ denote this Cauchy ...
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On the completeness of a metric space.
Theorem: Consider the metric space $(X, d)$, where $X \subseteq \mathbb{R}$ and $d(x, y) = |f(x) - f(y)|$, with $f$ being a one-to-one (injective) function on its domain $X$. If the range of $f$ is a ...
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Completeness of $(C^{\infty}(\mathbb{R}),d)$
Let $C^{\infty}(\mathbb{R})$ be the space of all infinitely differentiable complex-valued functions on $\mathbb{R}$. Define the metric
\begin{align}
d(f,g) = \sum_{n,m=0}^{\infty} 2^{-n-m}\frac{\sup_{...
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