Questions tagged [complete-spaces]

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

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Why complete space is good?

I wonder why the complete space is a good space. I know that for a metric space $X$, if any Cauchy sequence $\{x_n\}$ in $X$ converges, then $X$ is called complete metric space. But why such property ...
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Completeness of Metric Space

Consider the set $C^k(\mathbb{R^n},\mathbb{R^m})$ in the metric $\rho$ where $\rho$ is defined by $$\rho(f,g)=\sup\{d(f(x),g(x)),d(f^1(x),g^1(x)),\dots ,d(f^k(x),g^k(x))|x\in \mathbb{R^n}\}.$$ How can ...
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1answer
24 views

Completion of topological spaces.

Let be $Y \subset X$ be dense between rings with I-adic topology or between metric spaces. Can I conclude that the completion of $X$ is the same as the completion of $Y?$ I would think that it seems ...
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Homogeneous Besov space $\dot B^{-\sigma}(\mathbb R^d)$ is a Banach space

Let $\sigma > 0$ and $\theta \in S(\mathbb R^d)$ (Schwartz function) be fixed, where $\theta$ is such that its Fourier transform $\hat{\theta}$ is compactly supported, $0 \leq \hat{\theta} \leq 1$ ...
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2answers
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Proving a metric space is a complete metric space.

Let $X$ be a compact metric space, and let $B(X)$ be the set of real-valued bounded functions on $X$. For any $f, g ∈ B(X)$, define $$d_B(f, g) :=\sup _{x\in X}\left | f(x)-g(x) \right |$$ Suppose, we ...
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1answer
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Is it true that a metric over $\mathbb{R}^n$ (or $\mathbb{C}^n$) always makes a complete metric space?

I'm studying for my calculus II exam and I was thinking about the statement in the title. My "sloppy" proof is something like: Over $\mathbb{R}^n$ (or $\mathbb{C}^n$) the Bolzano-Weierstrass ...
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2answers
49 views

Completeness of a sequence space c(V)

I have an exercise given by: With $V \equiv \mathbb{R}^{n}$ being furnished with Euclidean rorm $\|\cdot\|_{2}$, define a normed space $$ c(V) \equiv\left\{\left(v_{j} \in V\right)_{j \in \mathbb{N}}:...
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1answer
28 views

$L^p + L^r$ is complete

From Folland, "Real Analysis, Modern Techniques and Their Applications", Section 6.1 exercise 4. Let $1 \le p < r \le \infty$. Show that $L^p + L^r$ with norm $\|f\| = \inf\{\|g\|_p + \|...
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1answer
37 views

$C^1$ manifold curve with bounded derivative has limit

Let $\mathcal{M}$ be a smooth Riemannian manifold (not necessarily complete) and consider $\gamma : [0,1) \to \mathcal{M}$ a differentiable curve with bounded derivative, i.e., there exists constant $...
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0answers
35 views

Every metric space $R$ has a unique completion such that $\phi(x)=x$ for $x\in R$. (Kolmogorov and Fomin)

I am reading a famous book by Kolmogorov and Fomin. Definition: Given a metric space $R$ with closure $[R]$, a complete metric space $R^*$ is called a completion of $R$ if $R\subset R^*$ and $[R]=R^*$...
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Is the closure of a set always complete?

Suppose we have a metric space $(X,d)$ that is not complete. (i.e. there are Cauchy sequences that do not converge). Would the metric space $(\bar{X}, d)$ (where $\bar{X}$ denotes the closure of $X$) ...
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1answer
57 views

About the definition of completion of a metric space $R$.

The definition of completion of a metric space $R$ is the following: Definition 1: Given a metric space $R$ with closure $[R]$, a complete metric space $R^*$ is called a completion of $R$ if $R\...
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1answer
30 views

Show that a set in a metric space is complete

Let X be the closed unit ball in $(\mathbb R ^2,||.||)$ where $||.||$ is the euclidean norm. Let : $$d(x,y)= \left \{ \begin{array}{r c l} ||x-y|| & \text{if x,y and 0 are on the same ...
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2answers
45 views

Prove that a metric space is complete.

Let $a,b \in [0,\infty)$ with $a \leq b$. $D(x,y)=\mid \frac{x}{2+x}-\frac{y}{2+y} \mid$. There exist constants $c_{1}, c_{2} \in [0,\infty)$ such that $$c_{1}\mid x - y \mid \leq D(x,y) \leq c_{2}\...
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Is the maximum norm on a vector space complete?

K is a complete non-archimedean field, and then K[x] can be seen as a vector space on K with infinite dimension. For f(x) in K[x], its norm is the maximum valuation the coefficients. Is this norm ...
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1answer
30 views

Do all Cauchy sequences in incomplete metric spaces have “natural limits” that simply lie outside the space?

The standard definition of a complete metric space is that all Cauchy sequences converge in the space. It's easy to come up with examples where a Cauchy sequence has a natural limit that simply is not ...
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2answers
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Completion of a seminormed space

Let $E$ be a normed space. A completion for $E$ is a pair $(\bar{E},\varphi)$ consisting of a Banach space $\bar{E}$ and a continuous linear map $\varphi:E\to \bar{E}$ that is injective, preserves ...
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1answer
44 views

Complete Probability Spaces

I am taking a course on Stochastic Analysis, and the introduction of the course covers Measure Theory\ Probability Theory. I am very new to this area of abstract maths and am trying to get to grips ...
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2answers
49 views

Property $d(x_n,x_{n+1})\le 10\cdot2^{-n}$ in a complete metric space

Let $(X,d)$ be a complete metric space and $(x_n)$ a sequence in $X$ such that $$d(x_n,x_{n+1})\le 10\cdot2^{-n}, \text{ for all $n$.}$$ Show that the sequence converges to some $a \in X$. Show that $...
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1answer
39 views

Space of bounded functions is complete

I've been trying to proof that the metric space $(\mathbf{B}(T,X), d_\infty)$ is complete, if $(X,d)$ is complete. The definitions are shown below: \begin{align*} \mathbf{B}(T,X) = \lbrace f: T \...
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Prova that $(\mathcal{S}(\mathbb{R^N}),d)$ is a complete metrical space.

Where $\mathcal{S}(\mathbb{R})$ is a Space of Rapidly Decreasing Functions, i. e., $f\in \mathcal{S}(\mathbb{R})$ if, only if, $f\in C^{\infty}(\mathbb{R})$ such that $$ \sup_{|\alpha|\leq j}\sup_{x\...
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4answers
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Let $E = \{(x,y) \in \mathbb{R}^2 \mid y=x^2 \}$. Show that $(E,d_E)$ is a complete metric space.

Let $E = \{(x,y) \in \mathbb{R}^2 \mid y=x^2 \}$. Show that $(E,d_E)$ is a complete metric space. Here $d_E$ is just the restriction of $d_2$ to $E$. I'm trying to show this using the preimage ...
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2answers
71 views

Is there a norm in which the vector space of all sequences with the induced metric is complete?

The question of knowing whether is possible put a norm in a vector space is something new to me. I liked the Ivo ideas and I have wondered a little more about a particular case. Is there a norm in ...
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1answer
41 views

Why is the unitization of $M_\infty(\mathbb{C})$ not complete?

Consider the space $M_\infty(\mathbb{C})$ of infinite matrices with only finitely many nonzero entries. This is a pre-C$^*$-algebra, and we can consider its unitization $M_\infty^1(\mathbb{C})$ (the ...
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Relationship between the notions of completeness for a metric space, a vector space and a field

My question is motivated by the varying notions of 'completeness' one attaches to these objects. Cauchy completeness: Pertaining to metric spaces. R with the Euclidean metric is Cauchy complete. LUB-...
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1answer
127 views

$X$ is complete iff $\sum_{n=1}^\infty \|x_n\| < \infty \implies \sum_{n=1}^\infty x_n$ converges (Carothers, Theorem $7.12$)

I have some questions to ask about the second part of the proof, i.e. $[\Leftarrow]$ direction: The author says, "As always, it is enough to find a subsequence of $(x_n)$ that converges". ...
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1answer
53 views

Complete/Connected space

The metric $d: \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R}$ defined by $$d(x, y) = \left\{ \begin{array}{lr} ||x||+||y|| & \mbox{if } x \neq y \\ 0 & \mbox{if } x = ...
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1answer
25 views

Convergence of the sequence $\frac{1}{nx}$ in $\mathbb{B}(0,1)$

Let $X=\mathbb{B}(0,1)$ denote the space of bounded functions from $(0, 1) \rightarrow \mathbb{R}$ with the supremum-metric. Does the sequence $(f_n)_{n=0}^\infty,f_n(x)=\frac{1}{nx}$ converge in $X$? ...
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0answers
61 views

$f(x)$ not satisfying Contraction Mapping Theorem but has fixed point

Topology without tears: I am asked to show using exercise 2 and 3 that $f:\mathbb{R}\to \mathbb{R}$ defined as $f(x)=\cos x$ does not satisfy the contraction mapping theorem but does have a fixed ...
4
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1answer
24 views

Prove that a set E is not complete by exhibiting a sequence of points in E that is Cauchy but does not converge to an element of E

Prove that the set $E=\{(x_1, . . . , x_d) :x_1, . . . , x_d>0\}$ in $\mathbb{R}^d$ is not complete by exhibiting a sequence of points $(x_n)_{n \in \mathbb{N}}$ in $E$ that is Cauchy but does not ...
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1answer
41 views

$C^1[0, 1]$ is complete with the following norm.

$C^1[0, 1]$ is complete with the $C_1$ norm. Consider the following norm, $$||f||=|f(0)|+||f^\prime||_\infty$$ This norm is equivalent to $C_1$ norm. But in the book they asked to prove the space is ...
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0answers
39 views

Graduate Real Analysis research based question- Completion of the real numbers

I am currently a graduate student taking a real analysis independent study class. This is my first time taking real analysis, as I did not take it as an undergraduate. I am working on a research ...
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0answers
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Completeness with Hilbert's metric

This picture was taken from this P. J. Bushell's paper. Here $E=\mathbb{S}^{n-1}\cap\mbox{int} K$, $M(x/y)=\max\{x_i/y_i\}_{i=1}^n$, $m(x/y)=\min\{x_i/y_i\}_{i=1}^n$ and $d(x,y)=\log\frac{M(x/y)}{m(x/...
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4answers
170 views

A countable compact set has infinitely many isolated points

I tried to prove that if $(X, d)$ is a complete metric space and $K$ a compact countable subset of $X$, then $K$ has infinite many isolated points. I tried to prove it with Baire's Theorem but I can't....
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2answers
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Why does every sequence of real numbers have an upper limit?

Let $\{a_n\}:\mathbb{N}\to\mathbb{R}$, and we define $E$ to be the set of all sub-sequential limits of $\{a_n\}$ as well as possibly the symbols $\infty$ and $-\infty$ if there are some sub-sequences ...
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1answer
46 views

Unitization of a $C^{*}$-algebra - completeness of the constructed norm

I have a question about the unitization of $C^{*}$-algebras. More precisely, a question about the proof of the following statement: If $A$ is a (possibly unital) $C^{*}$-algebra, then there is a ...
3
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1answer
77 views

Completeness Axiom of $\mathbb{R}$.

I use the following as the axiom of completeness of the reals $\mathbb{R}$: $$\forall X,Y\in \mathcal{P}(\mathbb{R})\backslash\{\emptyset\}: (\forall x\in X\quad\forall y\in Y: x\leq y) \implies \...
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3answers
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if two complete metric spaces coincide as sets, are their metrics (strongly) equivalent?

If two complete metric spaces coincide as sets, are their metrics (strongly) equivalent? If $p$ and $q$ are two strongly equivalent metrics (in the sense that $ A p(x,y) \leq q(x,y) \leq B p(x,y)$ for ...
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2answers
63 views

I want to prove completeness of $C[a,b]$ with metric induced by max norm, but need help

I added a picture, please check it. Can someone please explain the last three line of the proof? I thought by finding a cauchy sequence that converges to a continuous function we are done with proving ...
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1answer
64 views

Why we need complete space for the interchange of limits to be valid?

I am learning Real analysis using the book Analysis 2 by Terence Tao. I'm confused with the condition on which we can interchange the limit. Concerning the below proposition, my question is that: Why ...
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1answer
38 views

Check if $X$ is completely metrizable

Let: $$D_t = \{(x,y,t) \in \mathbb{R}^3 : x^2 +y^2 \leq 1\},$$ $$S=\{(p,q,0)\in \mathbb{R}^3: p^2 +q^2<1, p\in \mathbb{Q}, q\in \mathbb{Q} \}$$ $$I(a,b) \text{ - open line segment between points } ...
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1answer
57 views

The set $C= \{x \in X: \text{dist} (x,K) ≤\frac 12 \}$ is not compact

Give an example of a complete metric space $(X,d)$ and its compact subset $K$, that the set $C= \{x \in X: \text{dist} (x,K) ≤\frac 12 \}$ is not compact. From Heine-Borel theorem: completeness + ...
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1answer
17 views

Check the following statements.

Let $X=\{(x_1,x_2,\dots):x_i\in\Bbb{R}$ and finitely many $x_i$'s are non zero} and $d:X\times X\to\Bbb{R}$ be a metric on $X$ defined by $d(x,y)=\text{sup}_{i\in\Bbb{N}}|x_i-x_j|,x=(x_1,x_2,\dots),y=(...
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1answer
57 views

Prove that $(X,d)$ is complete

Let $(X,d)$ - metric space such that $A,B \subset X$ and $ X=A\cup B$ where $(A,d_{|A\times A}), (B,d_{|B\times B})$ are complete subspaces. Prove that $(X,d)$ is complete. I think it should be ...
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1answer
36 views

Banach contraction theorem with composition

Problem 5. Let $(\mathcal{X},\mathrm{d}_\mathcal{X})$ be a non-empty complete metric space. Suppose that $f,g:\mathcal{X}\longrightarrow\mathcal{X}$ are two Banach's contractions of $\mathcal{X}$. ...
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3answers
54 views

$\mathbb{Q}[\sqrt{2}]$ is not a complete space

I am trying to prove that $\mathbb{Q}[\sqrt{2}]=\{a+b\sqrt{2}:a,b\in\mathbb{Q}\}$ is not a complete space (with the standard metric). For that purpose, I am looking for a Cauchy sequence in $\mathbb{Q}...
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0answers
20 views

Is Space $(X, \bar{\rho})$ complete? ( What limit of sequence should I take?)

Let $X = ${$x\in \mathbb{R}^{\omega}$ , there exists $N \in \mathbb{N}$ such that $x_i$=0 for all $i\geq N$}, and let $\bar {\rho}$ be the uniform mertic on $ \mathbb{R}^{\omega}$ . Is (X, $\bar{\rho})...
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1answer
62 views

Lebesgue outer measure and complete metric space

Let $A$ be an elementary set. For $E,F\subseteq A$, let $E\sim F$ iff $E\triangle F$ is null. Define $d([E]_{\sim},[E']_{\sim})=m^*(E\triangle E')$. I've shown that $((\mathcal PA)/\sim,d)$ is a ...
3
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1answer
56 views

Space of Katetov-Functions with sup-metric is complete metric space

Let (X,d) be a metric space. $f: X\longrightarrow\mathbb{R}$ is called "Katetov map" iff : $$∀𝑥,𝑦,∈𝑋:|𝑓(𝑥)−𝑓(𝑦)|≤𝑑(𝑥,𝑦)≤𝑓(𝑥)+𝑓(𝑦)$$ The set of all Katetov-maps on X is denoted ...
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1answer
43 views

Complex and real matrix for eigenvalues

This is a short one but: Consider the real matrix $\alpha = \begin{pmatrix} 7 &3 &-4 \\ -2&-1 &2 \\ 6&2 &-3 \end{pmatrix}$ and let $\beta \in M_n (\mathbb{C})$ be the ...

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