Questions tagged [complete-spaces]
A metric space is complete if, in it, any Cauchy sequence is convergent.
746
questions
2
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3answers
84 views
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 ...
0
votes
0answers
47 views
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 ...
0
votes
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 ...
3
votes
0answers
74 views
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, $0 \leq \hat{\theta} \leq 1$ ...
0
votes
2answers
44 views
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 ...
-1
votes
1answer
26 views
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 ...
2
votes
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
$$
c(V) \equiv\left\{\left(v_{j} \in V\right)_{j \in \mathbb{N}}:...
2
votes
1answer
28 views
$L^p + L^r$ is complete
From Folland, "Real Analysis, Modern Techniques and Their Applications", Section 6.1 exercise 4.
Let $1 \le p < r \le \infty$. Show that $L^p + L^r$ with norm $\|f\| = \inf\{\|g\|_p + \|...
0
votes
1answer
37 views
$C^1$ manifold curve with bounded derivative has limit
Let $\mathcal{M}$ be a smooth Riemannian manifold (not necessarily complete) and consider $\gamma : [0,1) \to \mathcal{M}$ a differentiable curve with bounded derivative, i.e., there exists constant $...
3
votes
0answers
35 views
Every metric space $R$ has a unique completion such that $\phi(x)=x$ for $x\in R$. (Kolmogorov and Fomin)
I am reading a famous book by Kolmogorov and Fomin.
Definition:
Given a metric space $R$ with closure $[R]$, a complete metric space $R^*$ is called a completion of $R$ if $R\subset R^*$ and $[R]=R^*$...
0
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2answers
46 views
Is the closure of a set always complete?
Suppose we have a metric space $(X,d)$ that is not complete. (i.e. there are Cauchy sequences that do not converge). Would the metric space $(\bar{X}, d)$ (where $\bar{X}$ denotes the closure of $X$) ...
0
votes
1answer
57 views
About the definition of completion of a metric space $R$.
The definition of completion of a metric space $R$ is the following:
Definition 1:
Given a metric space $R$ with closure $[R]$, a complete metric space $R^*$ is called a completion of $R$ if $R\...
1
vote
1answer
30 views
Show that a set in a metric space is complete
Let X be the closed unit ball in $(\mathbb R ^2,||.||)$ where $||.||$ is the euclidean norm. Let :
$$d(x,y)= \left \{
\begin{array}{r c l}
||x-y|| & \text{if x,y and 0 are on the same ...
0
votes
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 $$c_{1}\mid x - y \mid \leq D(x,y) \leq c_{2}\...
0
votes
0answers
13 views
Is the maximum norm on a vector space complete?
K is a complete non-archimedean field, and then K[x] can be seen as a vector space on K with infinite dimension. For f(x) in K[x], its norm is the maximum valuation the coefficients. Is this norm ...
1
vote
1answer
30 views
Do all Cauchy sequences in incomplete metric spaces have “natural limits” that simply lie outside the space?
The standard definition of a complete metric space is that all Cauchy sequences converge in the space. It's easy to come up with examples where a Cauchy sequence has a natural limit that simply is not ...
2
votes
2answers
61 views
Completion of a seminormed space
Let $E$ be a normed space. A completion for $E$ is a pair $(\bar{E},\varphi)$ consisting of a Banach space $\bar{E}$ and a continuous linear map $\varphi:E\to \bar{E}$ that is injective, preserves ...
0
votes
1answer
44 views
Complete Probability Spaces
I am taking a course on Stochastic Analysis, and the introduction of the course covers Measure Theory\ Probability Theory. I am very new to this area of abstract maths and am trying to get to grips ...
1
vote
2answers
49 views
Property $d(x_n,x_{n+1})\le 10\cdot2^{-n}$ in a complete metric space
Let $(X,d)$ be a complete metric space and $(x_n)$ a sequence in $X$ such that $$d(x_n,x_{n+1})\le 10\cdot2^{-n}, \text{ for all $n$.}$$ Show that the sequence converges to some $a \in X$. Show that $...
0
votes
1answer
39 views
Space of bounded functions is complete
I've been trying to proof that the metric space $(\mathbf{B}(T,X), d_\infty)$ is complete, if $(X,d)$ is complete. The definitions are shown below:
\begin{align*}
\mathbf{B}(T,X) = \lbrace f: T \...
0
votes
0answers
34 views
Prova that $(\mathcal{S}(\mathbb{R^N}),d)$ is a complete metrical space.
Where $\mathcal{S}(\mathbb{R})$ is a Space of Rapidly Decreasing Functions, i. e., $f\in \mathcal{S}(\mathbb{R})$ if, only if, $f\in C^{\infty}(\mathbb{R})$ such that $$ \sup_{|\alpha|\leq j}\sup_{x\...
0
votes
4answers
41 views
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 ...
0
votes
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 ...
2
votes
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 ...
1
vote
0answers
31 views
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-...
3
votes
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". ...
0
votes
1answer
53 views
Complete/Connected space
The metric $d: \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R}$ defined by
$$d(x, y) = \left\{
\begin{array}{lr}
||x||+||y|| & \mbox{if } x \neq y \\
0 & \mbox{if } x = ...
0
votes
1answer
25 views
Convergence of the sequence $\frac{1}{nx}$ in $\mathbb{B}(0,1)$
Let $X=\mathbb{B}(0,1)$ denote the space of bounded functions from $(0, 1) \rightarrow \mathbb{R}$ with the supremum-metric. Does the sequence $(f_n)_{n=0}^\infty,f_n(x)=\frac{1}{nx}$ converge in $X$?
...
1
vote
0answers
61 views
$f(x)$ not satisfying Contraction Mapping Theorem but has fixed point
Topology without tears:
I am asked to show using exercise 2 and 3 that $f:\mathbb{R}\to \mathbb{R}$ defined as $f(x)=\cos x$ does not satisfy the contraction mapping theorem but does have a fixed ...
4
votes
1answer
24 views
Prove that a set E is not complete by exhibiting a sequence of points in E that is Cauchy but does not converge to an element of E
Prove that the set $E=\{(x_1, . . . , x_d) :x_1, . . . , x_d>0\}$ in $\mathbb{R}^d$ is not complete by exhibiting a sequence of points $(x_n)_{n \in \mathbb{N}}$ in $E$ that is Cauchy but does not ...
0
votes
1answer
41 views
$C^1[0, 1]$ is complete with the following norm.
$C^1[0, 1]$ is complete with the $C_1$ norm. Consider the following norm,
$$||f||=|f(0)|+||f^\prime||_\infty$$
This norm is equivalent to $C_1$ norm. But in the book they asked to prove the space is ...
1
vote
0answers
39 views
Graduate Real Analysis research based question- Completion of the real numbers
I am currently a graduate student taking a real analysis independent study class. This is my first time taking real analysis, as I did not take it as an undergraduate. I am working on a research ...
0
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0answers
22 views
Completeness with Hilbert's metric
This picture was taken from this P. J. Bushell's paper. Here $E=\mathbb{S}^{n-1}\cap\mbox{int} K$, $M(x/y)=\max\{x_i/y_i\}_{i=1}^n$, $m(x/y)=\min\{x_i/y_i\}_{i=1}^n$ and $d(x,y)=\log\frac{M(x/y)}{m(x/...
4
votes
4answers
170 views
A countable compact set has infinitely many isolated points
I tried to prove that if $(X, d)$ is a complete metric space and $K$ a compact countable subset of $X$, then $K$ has infinite many isolated points. I tried to prove it with Baire's Theorem but I can't....
0
votes
2answers
39 views
Why does every sequence of real numbers have an upper limit?
Let $\{a_n\}:\mathbb{N}\to\mathbb{R}$, and we define $E$ to be the set of all sub-sequential limits of $\{a_n\}$ as well as possibly the symbols $\infty$ and $-\infty$ if there are some sub-sequences ...
2
votes
1answer
46 views
Unitization of a $C^{*}$-algebra - completeness of the constructed norm
I have a question about the unitization of $C^{*}$-algebras. More precisely, a question about the proof of the following statement:
If $A$ is a (possibly unital) $C^{*}$-algebra, then there is a ...
3
votes
1answer
77 views
Completeness Axiom of $\mathbb{R}$.
I use the following as the axiom of completeness of the reals $\mathbb{R}$:
$$\forall X,Y\in \mathcal{P}(\mathbb{R})\backslash\{\emptyset\}: (\forall x\in X\quad\forall y\in Y: x\leq y) \implies \...
0
votes
3answers
32 views
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 ...
1
vote
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 ...
4
votes
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 ...
1
vote
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 } ...
2
votes
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 + ...
0
votes
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=(...
2
votes
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 ...
0
votes
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}$. ...
0
votes
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}...
0
votes
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})...
1
vote
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
votes
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 ...
2
votes
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 ...
