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Questions tagged [complete-spaces]

A metric space is complete if, in it, any Cauchy sequence is convergent.

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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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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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If $ \phi: X \to X^* $ is an isometry, then $ X $ is a complete space.

I am wondering if the following statement might hold (as I wanted to use this in solving another problem): If $ \phi: X \to X^* $ is an isometry, then $ X $ is a complete space. I know that $ X^* $ is ...
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Is the quaternion algebra an algebraically closed space/a complete metric space? [duplicate]

Let's consider the non commutative field of real quaternions. Is it algebraically closed, i.e. does every non constant polynomial with quaternion coefficients have a quaternionic root ? If seen as a ...
Mathias Richard's user avatar
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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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5 votes
2 answers
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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\...
Martin Sleziak's user avatar
3 votes
1 answer
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Completeness of $\sin(kx)_{k=0}^{\infty}$ in $L_1[1, 4]$

I am reading a function analysis book and having trouble proving the following task: is ${\sin(kx)}_{k=0}^{\infty}$ complete in $L_1[1, 4]$? I understand that completeness in $C[1, 4]$ implies ...
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3 votes
1 answer
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The space $C_c$ of real-valued compactly supported, continuous functions is not a Banach space under any norm

This answer showed the space $c_{00}$ of compactly supported sequences, is not a Banach space under any norm. I wonder if the same is true for the space $C_c$ of real-valued compactly supported, ...
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Assume that for every $f\in X^*$, there exists $y \in X$ such that $f(x)=\langle x, y \rangle$ for every $x \in X$. Show that $X$ is a complete space.

Let $(X, \langle \cdot, \cdot \rangle)$ be a real or complex vector space with an inner product. Assume that for every $f \in X^*$, there exists $y \in X$ such that $f(x) = \langle x, y \rangle$ for ...
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Rationals are incomplete and naturals are complete

Why are rationals incomplete and natural complete? I have been going through Analysis textbooks but I don't totally get the reason why. So, naturals are complete because you can divide them into two ...
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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 ...
Hilbert's Minion's user avatar
6 votes
2 answers
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Understanding the proof of $L^p(X,\mathscr{A},\mu)$ is complete ($1\leq p<+\infty$)

Background I have some questions when reading the proof of $L^p(X,\mathscr{A},\mu)$ is complete for $1\leq p<+\infty$. The proof is proceeded by showing that each absolutely convergent series in $L^...
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Cauchy but not convergent sequence on $C[-1,1]$

Consider $C[-1,1] = \{ f: [-1,1] \rightarrow \Bbb{R}: f$ is continuous $\}$, the inner product $\langle f,g \rangle = \int_{-1}^1 f(x)g(x) \, dx$ and $\lVert \cdot \rVert$ the induced norm. Then, the ...
Daniel García's user avatar
6 votes
1 answer
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Three topologies on the space of sections of a vector bundle

Let $E\to M$ be a Riemannian vector bundle over an oriented Riemannian manifold $(M,g)$ with a connection $\nabla$. Let $\Gamma(E)$ denote the vector space of sections of $E\to M$. For $\sigma \in \...
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Topological tensor product

Let $A$ and $B$ be topological abelian groups. Consider the following universal property for a pair $(P, \pi)$, where $P$ is a topological abelian group and $\pi: A \times B \to P$ is a jointly ...
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1 answer
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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 ...
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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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Inverse limits of complete metric spaces is Baire

It is well known that arbitrary products of complete metric spaces are Baire (refer to Dugundji, example). But, what happens when one considers inverse limits of complete metric spaces over an ...
mathable's user avatar
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Counterexample of Ekeland's variational principle

This is the version of the Ekeland's variational principle that i'm studying: Let $(X, d)$ be a complete metric space, and let $f: X \rightarrow \mathbb{R} \cup\{+\infty\}$ be a lower semicontinuous ...
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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 ...
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Is it possible to prove that $\mathcal{l}^2$ is a Hilbert space using the Riesz-Fisher theorem?

The space $\mathcal{l}^2$ of sequences of summable squares is famously known to be a Hilbert space when given the scalar product: $$\langle (u_n)_{n\in\mathbb{N}},(v_n)_{n\in\mathbb{N}} \rangle:= \...
IAG's user avatar
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Uniqueness of extension of the normed space structure to a larger topological space

The following popped up when I was contemplating the completion of normed linear spaces. Question: Let $E$ be an NLS (normed linear space) and $X$ a topological space. Let $E$ sit inside $X$ as a ...
Atom's user avatar
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Completes of an orthonormal set in a Hilbert space

Let numbers $\{\rho_n\}_{n\ge 0}, \quad \rho_n \neq \rho_k,\quad (n\neq k)\quad$ of the form $\quad \rho_n = n + \frac{a}{n}+\frac{\kappa_n}{n}, \quad \{\kappa_n\} \in \ell^2 \quad $ be given. Then $$ ...
Juan's user avatar
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1 answer
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Is the Axiom of Completeness logically equivalent to "There is no proper superset of $\mathbb R$ that is an ordered Archimedean field"?

The Axiom of Completeness can be formulated as: There exists a set $R$ such that: $R$ is an ordered Archimedean field Any nonempty subset of $R$ with an upper bound has a least upper bound. Recently,...
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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 ...
Michael Wang-Wakamatsu's user avatar
2 votes
2 answers
154 views

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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1 answer
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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 ...
Atom's user avatar
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Applying Hilbert's geometric Axiom of Completeness to deduce equivalent forms

In Hilbert's Foundations of Geometry, the axiom of completeness is presented as follows: To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner ...
Circuit Craft's user avatar
2 votes
2 answers
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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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1 vote
1 answer
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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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1 answer
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Find the completion of a metric space [closed]

Prove that $\mathbb{R}$ with metric $\rho(x,y)=|\arctan x-\arctan y|$ is not a complete metric space. Determine its completion. Here is the link to prove the given metric is not complete. Here is the ...
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1 vote
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Topology of completion of abelian first-countable topological group

Let $G$ be an abelian first-countable topological group. Define $CS(G) := \{(x_n) \in G^{\mathbb{N}} \ | \ (x_n) \subset G \text{ is Cauchy-sequence}\}$ and $NS(G):=\{(x_n) \in CS(G) \ | \ (x_n) \...
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Mathematical induction and completness of metric space

Mathematical induction in form of - "in well-founded poset any progressive subset is total" (by progressive of $S$ i mean that $\forall x \ ((\forall a<x \ (a \in S)) \rightarrow x \in S)$...
nagvalhm's user avatar
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4 answers
118 views

$X = (0, +\infty)$. Function $d(x, y) = \left|\frac{1}{x} - \frac{1}{y}\right|$. Is $(X, d)$ complete? [duplicate]

I'm having trouble understanding and solving this problem. For $(X,d)$ to be complete, every Cauchy sequence of points in $X$ has to have a limit in $X$. Let's take a sequence $(x_n)$ in $X$. For ...
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1 answer
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$(M, g)$ complete Riemannian manifold implies $(M, \tilde{g})$ complete if $\|X\|_{\tilde{g}} \ge \|X\|_g$ for all $X$

The following statement appears as a remark in "An Analytic Criterion for the Completeness of Riemannian Manifolds" by William Gordon (1973): Let $g$ and $\tilde{g}$ be two metric tensors ...
infinitylord's user avatar
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A restricted product of lp spaces

For each $p \in \mathbb{R}$, we consider $l_p (\mathbb{Z}) := \{ f : \mathbb{Z} \rightarrow \mathbb{R} : \sum_{i = 0}^{\infty} |f(i)^p| < \infty \}$. I am trying to figure out how a certain ...
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2 votes
1 answer
73 views

Show that $X=(-1,1)$ with $d(x,y)=\left|\frac{x}{1-x^2} - \frac{y}{1-y^2}\right|$ is a complete metric space.

On the space $X = (-1,1)$ with $d(x,y)=\left|\frac{x}{1-x^2} - \frac{y}{1-y^2}\right|$. Prove that $(X,d)$ is complete metric space. What I've tried: I considered a Cauchy sequence $(x_n)_{n \in \...
VarúAnselmo Sui's user avatar
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Countable family of norms and seminorms

In volume 4 of Gelfand and Vilenkin they study countably normed spaces. I assume a definition for a countably normed space was given in an earlier volume, but I do not have access to them and I could ...
CBBAM's user avatar
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Question about completely metrizable spaces

We say a metric space X is completely metrizable if there a metric that induces its usual topology and X with that metric is complete. I´ve seen, for example, that (0,1) is completely metrizable and ...
Albi's user avatar
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1 vote
1 answer
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Discontinuity points of a function over dense topologically complete subset.

I spend a lot time with this problem, and I need your help, Let $(X,d)$ be a metric space and $A$ be a dense subset of $X$, suppose that there is a bounded distance $d’$ on $A$ topologically ...
Ifielmodes's user avatar
4 votes
1 answer
85 views

Find a Closed and Bounded set but not Compact in an incomplete metric space

I am learning topology by myself: Let $(X,d)$ be an incomplete metric space. Show that there exists a closed and bounded set $E \subset X$ that is not compact. My attempt: Since $X$ is incomplete ...
Blue Tea's user avatar
2 votes
1 answer
73 views

Let $(X,d)$ and $(Y,\rho)$ be two metric spaces with a bijection between them. If $(X,d)$ is complete then can we say $(Y,\rho)$ also complete?

The question arise when I study complete space . We know if there a bijection between two set then their property will same ,is it true for completeness property of a metric space. We know $\mathbb{R^...
ëlêtro's user avatar
1 vote
1 answer
26 views

Finite index dense normal subgroups of completely metrizable groups

Is there some completely metrizable group $M$, which contains a normal subgroup $N\trianglelefteq M$ of finite index (at least $2$) that is dense in $M$?
user12345's user avatar
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0 answers
34 views

completeness of real and convergence of binary series

In page 'Completeness of the real numbers' of Wikipedia is said that: '' completeness (of real) is equivalent to the statement that any infinite string of decimal digits is actually a decimal ...
user791759's user avatar
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0 answers
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Open normal subgroups with finite image under extensions

Let $A$ be a (discrete) countable group such that for ever completely metrizable group $M$ and any (not necessarily continuous) homomorphism $f\colon M \to A$ there exists some open normal subgroup $N\...
user12345's user avatar
1 vote
0 answers
57 views

If Sobolev space is isometric to $L^2(\mathbb{R})$ then Sobolev space is complete?

My attempt: Let $H^2(\mathbb{R})=\left\{u\in L^2(\mathbb{R}): \left\|u\right\|_{H^2(\mathbb{R})}:=\left\|\mathcal{F}^{-1}((1+\xi^2)\widehat{u})\right\|_{L^2(\mathbb{R})}<\infty\right\}$ the Sobolev ...
eraldcoil's user avatar
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1 vote
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Are time-dependent Banach space-valued functions a Frechét space?

Consider a family of Banach spaces $(\mathcal{B}_{t},\Vert\cdot\Vert_{t})_{t\in\mathbb{R}}$ and define the space $C^{\infty}(\mathbb{R},\mathcal{B}_{\bullet})$ where smootheness has to be understood ...
B.Hueber's user avatar
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1 vote
0 answers
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A question for uniform convergent of sequence of functions.

Consider the sequence of functions $<f_n(t)>$, defined as $ f_n(t) = \begin{cases} e^{-t^2} & \text{if } -n \leq t \leq n \\ \frac{e^{-n^2}}{[1-n(t-n)]} & \text{if } n \leq t &...
neelkanth's user avatar
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