Questions tagged [complete-spaces]

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

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2answers
122 views

Complete Metric Space can be a Banach Space?

Let $(S,d)$ be the space of all sequences in $\mathbb{R}$ with the metric $$d(\mathbf{x},\mathbf{y})=\sum_{i=1}^{\infty}\dfrac{1}{2^i}\dfrac{|\xi_i-\eta_i|}{1+|\xi_i-\eta_i|}$$ where $\mathbf{x}=(\...
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1answer
36 views

How do we prove that compact spaces in metric spaces are bounded?

Let $(X,d)$ be a compact metric space. Then $(X,d)$ is both complete and bounded. My solution The space $(X,d)$ is indeed complete. This is because every Cauchy sequence which admits a convergent ...
3
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1answer
29 views

Prove that $Y$ is complete iff it is closed.

Let $X$ be a complete metric space and $Y$ be a subspace of $X$. Prove that $Y$ is complete iff it is closed. My attempt: Firstly, suppose that $Y$ is closed. We want to show that $Y$ is complete. ...
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1answer
30 views

Why is the Gromov-Hausdorff metric complete?

I've been reading up on the Gromov-Hausdorff metric, using the following books: A Course in Metric Geometry, by Dmitri Burago, Yuri Burago, and Sergei Ivanov Metric Structures for Riemannian and Non-...
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1answer
15 views

Closure of a sigma algebra and complete measure spaces

Suppose $<X,M,\mu>$ is a measure space and $\overline{M}=\{A:\exists B,C \in M, B\subset A \subset C\ \text{and}\ \mu(C \smallsetminus B)=0\}$ is the closure of the $\sigma$-algebra $M$. I'm ...
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1answer
36 views

Space of continuous functions form a Banach space?

Let $X$ be the collection of all continuous real-valued functions defined by \begin{equation*} ||f||=\sup_{x\neq y}\frac{\left\vert f(x)-f(y)\right\vert }{\left\vert x-y\right\vert } \end{equation*} ...
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0answers
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Completion of a norm space, proof understanding.

I just came across a proof that shows that each normed spaces can be completed to a Banach space. I would love to get some things straightned out though. This is the proof: https://prnt.sc/shl2f4 ...
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1answer
43 views

Show that $C([0,1],\mathbb{R}^{2})$ is complete.

I need to show that $C([0,1],\mathbb{R}^{2})$ is complete. $C([0,1],\mathbb{R}^{2})$ are continuous functions $f:[0,1]\to\mathbb{R}^{2}$ and $\;d(f,g)=\sup_{x\in[0,1]}\left\Vert f(x)-g(x)\right\Vert ...
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2answers
45 views

Showing incompleteness of metric space by specifying non-convergent Cauchy sequence

I have been looking at the space of continuous functions over a compact interval $C([0,2])$ equiped with the the integral norm of absolut values $\| \cdot \|_1$. I read a counterexample that showed ...
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1answer
20 views

Let $(Y,d|_{Y\times Y})$ be a subspace of $(X,d)$. If $(Y, d|_{Y\times Y})$ is complete, then $Y$ must be closed in $X$.

(a) Let $(X,d)$ be a metric space, and let $(Y,d|_{Y\times Y})$ be a subspace of $(X,d)$. If $(Y, d|_{Y\times Y})$ is complete, then $Y$ must be closed in $X$. (b) Suppose that $(X,d)$ is a complete ...
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1answer
13 views

Let $(X,d)$ be a complete metric space and suppose that $A$ and B are both dense and $G_δ$ subsets of X, show that $A\cap B\neq \emptyset$.

A set is said to be $G_\delta$ set if it can be expressed as countable intersection of open sets. Let us suppose on the contrary that $A\cap B=\emptyset$ Now, choose $x\in B$. Then there exists a seq $...
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2answers
31 views

Is the delta neighborhood of an epsilon neighborhood of a complete metric space the same as the delta + epsilon neighborhood

Let X be a complete metric space. Does there exist a compact subset A that is non empty s.t. the delta neighborhood of the ...
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1answer
29 views

Complete Metric Space Proof

Fix $T \ge 0$ and let $\mathcal{R}_c^2$ be the set of continuous adapted stochastic processes $X$ such that $\|\sup_{t\leq T} |X_t| \|_{L^2} < \infty$ with metric $$d(X,Y) = \|\sup_{t \leq T} |X_t -...
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0answers
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The completion of $C_c(\mathbb{R}^k)$ with $\|f - g\|_p$ as a metric

In Rudin RCA, 3.15, he remarks that because $C_c(\mathbb{R}^k)$ is dense in $L^p(\mathbb{R}^k)$ (i.e. every point of $L^p(\mathbb{R}^k)$ is either in or a limit point of $C_c(\mathbb{R}^k)$) and ...
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0answers
21 views

Properties of equivalence metrics

Firstly, I want to share some properties I found for uniformly equivalent metrics. Suppose $(X,d)$ and $(X,p)$ are uniformly equivalent, then the identity map and its inverse are uniformly continuous. ...
4
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1answer
64 views

Determine if $X=\{f\in C^1[0,1] | f(0)=f'(0)\}$ is complete WRT $||.||_{\infty}$ norm and show that $X$ is infinite dimensional.

I'm having trouble determining if $X$ is complete WRT $||.||_{\infty}$ norm. I know that in order to show that I need to take a Cauchy sequence and show that it has a limit in my space $X$ or find a ...
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0answers
16 views

Prove that X is Banach space [duplicate]

Let $X :=C^0_{\mathbb{C}}[0,1]=\left\{f:[0,1]\mapsto \mathbb{C}; f\in C^0\right\}$. In order to prove $X$ is a Banach space, we need to prove that is a complete normed vector space. Thus, we will use ...
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0answers
11 views

Function from metric space to complete metric space [duplicate]

"Let $(X, d_X)$ be any metric space, and $(Y,d_Y)$ be a complete metric space. Prove, that space $C(X, Y)$ of all continuous and limited functions $f : (X, d_X) \rightarrow (Y, d_Y)$ with supremum ...
0
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1answer
30 views

A counterpart of Bolzano-Weierstass theorem in metric space.

In $\mathbb R$,every bounded infinite set $A\subset \mathbb R$,has a limit point in $\mathbb R$.But if we replace $\mathbb R$,by any other metric space $X$,then this may not hold.For example $X=\...
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0answers
87 views

Can Hilbert spaces be defined over fields other than $\mathbb R$ and $\mathbb C$?

Let $V$ be a vector space over a field $K$. Suppose $K$ has an involution $*:K\to K,z\mapsto z^*$, meaning $*$ is an automorphism and that $(z^*)^*=z$ for all $z\in K$. Define a $K$-inner product ...
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1answer
46 views

The Completion of a Measure Space. [closed]

Let $(X, \mathcal{T}, \mu)$ be a measure space and $$\mathcal{N} = \{N\subset X: \exists A\in \mathcal{T}\ \textrm{such that}\ N\subset A\wedge \mu(A)=0\}.$$ Let's define $$\mathcal{T}_1 = \{B\cup N: ...
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1answer
34 views

Is $\mathbb{N}$ a complete metric space with this metric $d(a,b) = \sqrt{1-2\frac{\gcd(a,b)}{a+b}}$?

Is $X:=\mathbb{N}$ a complete metric space with this metric $d(a,b) = \sqrt{1-2\frac{\gcd(a,b)}{a+b}}$? Thanks for your help! Edit: This metric plays a role in the formulation of the abc-conjecture: ...
0
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2answers
34 views

Prove that metric space of polynomial is not complete

Prove that the metric space $P[a,b]$ of all polynomials with uniform metric $d(f,g) = \max|f(t)- g(t)|$ for every $f, g$ belonging to $P[a,b]$ and $t$ belonging to $[a,b]$ is NOT complete. My ...
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0answers
22 views

Is there a complete non barrelled locally convex space?

These answers provides two examples of non barrelled locally convex spaces, but they are both incomplete. Furthermore, their completion is a Banach space and so necessarily barrelled. This answer ...
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0answers
19 views

Showing this space is complete [duplicate]

Let $(c_0,\|\space.\|_{\infty})$ be the space of sequences $(x_k)_{k=0}^{\infty}$ such that $x_k\in{}\mathbb{K}$ and $x_k\rightarrow{}0$. I want to show that this space is complete. I understand this ...
2
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1answer
39 views

Prove that $f^{-1}$ is uniformly continuous

Suppose $f:X \rightarrow Y$ bijective, $f$ is uniformly continuous, $f^{-1}$ continuous, $X, Y$ are complete. It is required to prove, that $f^{-1}$ is uniformly continuous as well. My idea was to ...
0
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1answer
22 views

Difference between sequential criteria for total boundedness (Carothers Lemma 7.3 and Theorem 7.5)

Carothers (p. 90-91) includes the following lemma and theorem when describing the sequential criterion for total boundedness. I'm having trouble understanding how they are different. Could someone ...
1
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1answer
69 views

Proof Completeness of a Metric Space with Complex Measure

So I have a lot of trouble with this problem: Let $M(X)$ be the set of signed measures on $(X, \mathcal A)$. We define a metric by $d(\mu_1,\mu_2)=|\mu_1-\mu_2|(X)$. Proof that the metric space ...
1
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0answers
45 views

Virtual point of a metric space.

For a metric space $X$, consider a fixed point $z$. Let $\delta_z(x) = d(z,x)$ denote the distance from a point $x$ to $z$. Then $\delta_z$ (or, more simply $\delta$) is a function from $X$ to $\...
0
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0answers
13 views

Confusion with nomenclature of Cantor intersection property.

Suppose $X$ is a metric space.Then $X$ is complete iff For each sequence of non-empty closed sets viz. $(F_n)$such that $F_{n+1}\subset F_n$ and $\operatorname {diam}(F_n)\to 0$,$\cap_{n=1}^{\infty} ...
2
votes
1answer
42 views

Completeness of the dual space of a Frechet Space

I'm having trouble understanding completeness of duals of Fréchet spaces, I have been reading Meise and Vogt "Introduction to Functional Analysis" and something is not clear to me. So let $\mathcal{F}$...
0
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0answers
11 views

How to prove that a recursively defined contraction mapping on a sequence is convergent

I'm asked to prove the following statement: Let $f$ be a contraction mapping on a complete metric space $M$ in the sense that $$ d(f(x), f(y)) \leq c d(x, y), \quad \forall x, y \in M $$ for some $c \...
2
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1answer
30 views

$\sup L^1$ space with uniform integrability

I have a following question. For $t\in[0,T]$, let $f_t:\mathbb{R}\rightarrow\mathbb{R}$ be uniformly integrable family i.e. $\{f_t,\,t\in[0,T]\}$- uniformly integrable. We consider the space of such ...
3
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0answers
39 views

Completeness of Metric in $C^1$

Consider $C^1[0,1]$ with the metric $d(f,g)=d_{\infty}(f,g)+d_{\infty}(f',g')$. Then $C^1$ is complete. My attempt: Let $(f_n)_n$ be cauchy in $C^1[0,1]$ wrt $d$. Then $f_n$ and $f'_n$ are cauchy ...
0
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0answers
47 views

Space of continuous functions $f: [a,b] \to G$ where $G \subset \mathbb{R}^{d}$ is closed

I'm trying to prove the following claim in my lecture notes on ODEs. Theorem: Let $f: [a,b] \to G$ where $G \subset \mathbb{R}^{d}$ is closed and define $\| f \|_{\infty}=sup_{t}\lvert ...
1
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1answer
47 views

Is $d(f,g)=\int_{0}^{1} f(x)g(x)$ a metric for $P([0,1])$?

I think it is not a metric for the space because if you take $f(x)=x$ a polynomial in $P([0,1])$ then I can show that: $d(x,x) = \int_{0}^{1} x^{2}dx=\frac{x^{3}}{3} \big|_{0}^{1}=\frac{1}{3} \neq 0$ ...
1
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1answer
44 views

Show that this metric space isn't complete

Consider the space $X$ of all continuously differentiable functions from $[0,1]$ to $\mathbb{R}^{2}$, such that $f(0) = a$ and $f(1) = b$, where $a,b \in \mathbb{R}^{2}$ and $f \in X$. Define the ...
0
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0answers
40 views

French Railroad metric complete? [duplicate]

Let $(X,d)$ be any metric space and let $p\in X$ and let $d_p$ be the corresponding french railroad metric. That is, $d_p(x,y)=d(x,p)+d(p,y)$. I'm supposed to show that $(X,d_p)$ is necessarily a ...
1
vote
1answer
39 views

Are $I$-adically complete $R$-modules the same as $I$-adically complete $R^\wedge_I$-modules?

Let $R$ be a commutative ring and $I \subseteq R$ an ideal. Let $R^\wedge_I$ be the $I$-adic completion of $R$. Let $Mod_{R^\wedge_I}^{comp}$ be the category of complete $R^\wedge_I$-modules and ...
0
votes
1answer
48 views

Equivalence of Baire Theorem

In my studies of functional analysis I have come across this two statements of the Baire Category Theorem. Let $(X,d)$ be a complete metric space. Let $U_n$ be a dense, open set for each $n \in \...
2
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2answers
57 views

The space $(\mathbb{Q}_{p}[X],|-|_{\text{gauss}})$ is not complete

I'm stuck trying to prove incompletness of certan $p$-adic space. Let $\mathbb{Q}_{p}[X]$ the space of polynomials. We know that has infinite dimension. On $\mathbb{Q}_{p}[X]$ define the function $\...
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1answer
36 views

Confusion about the proof of uniform convergence

Let $Y$ be a complete metric space. Then a uniformly Cauchy sequence $(f_n)$ of functions $f_n$ : $X → Y$ converges uniformly to a function $f : X → Y $. I met this proof in the book, and I have ...
2
votes
1answer
54 views

Can I determine the finite subcover of any given open cover?(An unknown side of a known result)

Heine-Borel theorem in $\mathbb R$ states that for a closed,bounded set,we can get a finite subcover for every given open cover.But the proof is existential and uses completeness(in the form of Cantor-...
-1
votes
1answer
15 views

Clarification about isometric imbedding of a metric space into its completion

I can't understand the conclusion of the proof of theorem 43.7 in Munkres' Topology. We consider a metric space $(X, d)$ and a fixed pont $x_0 \in X$. He defines an imbedding $h : X \to \mathcal{B}(...
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votes
1answer
36 views

Is the metric space of polynomial of degree $\leq1$ complete with the following metric?

I must determine if the following metric space is complete $$\mathbb R_1[x]\times \mathbb R_1[x]\rightarrow \mathbb R: d(p(x),q(x))=max\{|p(0)-q(0)|,|p(1)-q(1)|\}$$ where $$\mathbb R_1[x]=\{p(...
0
votes
1answer
35 views

completeness of the metric $[1,2]\times[1,2]\rightarrow \mathbb R:$ $d(x,y)=|x^4-y^4|$

I must determine if the following space is complete $[1,2]\times[1,2]\rightarrow \mathbb R:$ $d(x,y)=|x^4-y^4|$. Is $([1,2],d) $complete? my try: Let $\{x_n\}$ be a Cauchy sequence by ...
1
vote
1answer
42 views

What is wrong with my solution about the completeness of this metric space?

For $x_1,x_2 \in X = (1, 3) \subset \mathbb R$ it is defined a metric $ d(x_1, x_2) = |\frac{1}{x_1}-\frac{1}{x_2}|$ I must determine if $(X; d)$ is a complete metric space The solution I was ...
0
votes
1answer
30 views

Tell if a set $K \subseteq\mathbb R $, closed in $\tau_d$ and bounded by the distance d is necessarily compact in $\tau_d$

Considering the following metric over $\mathbb R$: $d(x,y)=|x-y|/(1+|x-y|)$ I have to 1) find if ($\mathbb R$,d) is a complete metric space 2)Tell if a set $K \subseteq\mathbb R $, closed in ...
0
votes
1answer
108 views

Completeness of $L^2([0,1]) \otimes L^2([0,1])$.

I'm trying to understand the Hilbert space of a quantum system of two particles in the infinite square well. The claim is that this is $L^2([0,1] × [0,1])$, which is said to be isomorphic to the ...
0
votes
1answer
72 views

Equivalent notion of completion of metric space.

In my introductory Topology course, a completion of a metric space $(X,d)$ is a complete metric space $(\tilde{X}, \tilde{d})$ together with an isometry $i : (X, d) \rightarrow (\tilde{X}, \tilde{d})$ ...

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