Questions tagged [complete-spaces]
A metric space is complete if, in it, any Cauchy sequence is convergent.
0
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0answers
21 views
When can an isometry be constructed between to metric spaces?
I wanted to show that the space $\mathcal{S} := \{ x = (x_k)_{k \in \mathbb{N}} \subset \mathbb{C}\}$ with the metric
\begin{equation*}
d: \mathcal{S} \times \mathcal{S} \to \mathbb{R}, \
(x,y) \...
0
votes
1answer
28 views
$\ell^p(J)$ is complete (Banach)
I have to prove that the space
$\ell^p(J)$ defined as the set of all functions
$\psi: J\rightarrow \mathbb{F}$
s.t. $\psi$ is null except in a contable subset of $J$
and
$||\psi||_p :=\bigg(\...
0
votes
1answer
44 views
Prove that if metric spaces $(X, d_{X})$ and $(Y,d_{Y})$ are complete so is metric space $X\times Y$ complete
a) Metric spaces $(X, d_{X})$ and $(Y,d_{Y})$ are complete. Prove that metric space $X\times Y$ is complete if the metric is defined as:
$\rho((x_1,y_1),(x_2,y_2))$ = $\sqrt{d_X(x_1,x_2)^2 + d_Y(...
0
votes
0answers
15 views
Is the space of all infinitesimal numeric sequences complete?
Is the space $c_0$ of all infinitesimal numeric sequences $a = (a_1, a_2,...,a_n,...)$ complete? The norm is $||a|| = sup|a_i|$.
As i couldn't prove that every Cauchy sequence has a limit in $c_0$ (...
7
votes
1answer
180 views
Ultralimit of metric spaces is complete
Consider the following short proof that all ultralimits are complete
(Thy typo $d_omega$ is of course to be read as $d_\omega$.)
There are two things that I dont understand about this proof.
1) Why ...
1
vote
0answers
29 views
Bochner integral on non complete space
Let $X_0=(X,\|\cdot\|_0)$ be a Banach space and $X_1=(X,\|\cdot\|_1)$ a normed vector space (not complete) such that there is a constant $c>0$ such that
\begin{align*}
\|\cdot\|_1 \leq c\|\cdot\|_0
...
1
vote
1answer
27 views
Counterexample for $(\mathcal{B}([a,b], Y), d^{\sup})$ complete $\implies Y$ complete
Theorem: Let $(Y,d)$ be a complete metric space. Then the metric space $(\mathcal{B}([a,b], Y), d^{\sup})$ is complete, where $B(C,D)$ is the set of all bounded functions form $C$ to $D$ and $d^{\sup}(...
0
votes
1answer
21 views
Maybe “misleading” use of $\liminf$ in proof of $L^{\infty}$ completeness (from Linear Functional Analysis by Hans Wilhelm Alt, p.52)
I have come across suspicious use of liminf a few times, while reading the book. So I decided to post one short case because it seems to me I am probably understanding it wrong. For me limsup does ...
0
votes
2answers
26 views
Is there a metric for which the open unit interval is complete?
Let, $I= (0,1)$ It is well known that $I$ is not a complete with respect to the Euclidean metric $(x,y)\mapsto |x-y|$.
However, $(I,|\cdot|)$ is separable.
Question: Can we find a metric $d: I\...
2
votes
1answer
15 views
Completeness of a Normed Space of Smooth, Bounded Functions
As part of a proof of the Picard–Lindelöf theorem, I am using the following space:
$X = \{ u \in C([0,T]) : u(0) = \alpha , || u - \alpha || \leq K\}$
where $K \in \mathbb{R}_{> 0} , \ \alpha \in ...
0
votes
1answer
19 views
To check Completeness and totally boundedness of a metric space.
Let$X=N$ be set of positive integers.Consider the metric d such that.
$d(m,n)=\frac{1}{m}-\frac{1}{n}$.Then how to verify whether $(X,d)$ is complete and totally bounded?
We know that only Cauchy ...
2
votes
1answer
22 views
Proof that the space $L^1$ is complete in its metric
I don't understand a particular detail of the proof on page 70 of Stein and Shakarchi's Princeton Lectures in Analysis III.
We consider a Cauchy sequence $\{f_n\}$ in $L^1$ and choose a subsequence $...
1
vote
1answer
31 views
Proving that the metric space of all sequences of positive integers is complete
Consider the set of all sequences of positive integers with the following metric:given $x=(n_j)$, $y=(m_j)$
$$d(x,y)= 1/\inf\{j: n_j \ne m_j\}$$ if $x\ne y$ and $0$ otherwise.
I want to show that it ...
1
vote
2answers
27 views
Complete metric space on unique metric
Consider the metric $$d(m,n) = \frac{|m-n|}{mn}$$
Is this metric on the natural numbers $(1,2,\ldots)$ complete?
I'm struggling but heres an idea I have from reading other similar questions.
The ...
2
votes
2answers
41 views
How do I prove completeness / incompleteness on a set of functions?
I'm asked to show that
$$((C[0,1], \mathbb{R}), ||f||_1), \quad ||f||_1 = \int_0^1{|f(x)|dx}$$
is not a Banach space.
$||f||_1$ is a norm on this space so therefore I need to show that the space ...
0
votes
0answers
24 views
Regression through linear Fourier coefficient fitting?
Basically suppose on was given an unknown function/data and expected to write a function so that $Y=f(X)$, this can be done by linear regression in simple cases very easily. However, suppose that the ...
1
vote
2answers
51 views
Proof that the metric space $M$ is complete if every closed ball of $M$ is complete.
Let $M$ be a metric space
I'm asked to prove the statement
"Every closed ball of $M$ is complete $\implies$ $M$ is complete".
My attempt at this is as follows:
Let $\{y_i\}$ be a cauchy ...
0
votes
0answers
22 views
Unit closed Ball in completion
Let $(E, \|\cdot\|_E)$ be any normed vector space. Consider the canonical identification with its double dual $(E'', \|\cdot\|_{E''})$ given by
$$i_E: (E, \|\cdot\|_E) \to (E'', \|\cdot\|_{E''}), \; \...
0
votes
0answers
22 views
The completeness of vector spaces $\mathbb{R}^{d}$ and $\mathbb{C}^{d}$ [duplicate]
I trying to prove that the vector spaces $\mathbb{R}^{d}$ and $\mathbb{C}^{d}$ are complete, which means that every Cauchy sequence in the norm converges to a limit in vector space.
$\mathbb{R}^{d}$ ...
0
votes
1answer
169 views
The space of all finite-degree polynomials $\mathbb{P}$ is not complete in any norm. [closed]
Let $\mathbb{P}$ denote the space of all finite degree polynomials in one
variable. Show that $\mathbb{P}$ is never complete with respect to $P_1$ norm, i.e.,
$\|\cdot\|_1$ by giving an example ...
0
votes
0answers
48 views
Does there exist an intermediate step of completion of the algebraic closure $\bar{\mathbb{R}}$ to get the field $\mathbb{C}$?
We know that completion of the field of rationals $\mathbb{Q}$ is the field $\mathbb{R}$ of reals with respect to usual metric.
Next, algebraic closure of $\mathbb{R}$ is the field $\mathbb{C}$ of ...
0
votes
0answers
30 views
Completeness of the tangent bundle of riemannian manifold
Let $(M,g)$ be a riemannian manifold and $TM$ its tangent bundle. There are natural riemannian metrics that we can endow the tangent bundle with (for instance the Sasaki metric) and I wonder if for ...
1
vote
1answer
74 views
Normed space $C^2[0,1]$ with norm $\lVert f\rVert:=\max_{t\in[0,1]}\{\lvert f(t)\rvert+\lvert f''(t)\rvert\}$ is Banach space
The problem is as follows: I want to show that the normed space $C^2[0,1]$ with norm defined as $$\lVert f\rVert:=\max_{t\in[0,1]}\{\lvert f(t)\rvert+\lvert f''(t)\rvert\}$$ is a Banach space (and I ...
1
vote
1answer
40 views
completeness definition (real analysis basic) (many edits)
I'm trying to get a basic definition of completeness in my head (as we are not using the cauchy/limit defn in class). The basic sense that I get, is that a set is complete if there are no 'holes' in ...
0
votes
1answer
29 views
$c_{00}$ is not complete
I try to show that the space $c_{00}=\{(x_n):x_n=0 \text{ all but finitely many }n\}$ is not complete with respect to the norm $\|x\|_\infty=\max |x_n|$.
My attempt:
Let $(z_n)=\left(1,\frac{1}{2},\...
2
votes
2answers
107 views
Show that the class $C_c(\mathbb{R^n})$ of continuous functions with compact support is not a complete metric space
I'm asked to show the following:
Show that the class $C_c(\mathbb{R^n})$ of continuous functions with compact support on $\mathbb{R^n}$ with the sup-norm metric $d(f,g):= \text{sup}_{x\in \mathbb{R^...
2
votes
0answers
72 views
Critique my proof: Proving that $C[a,b]$ is not a Hilbert Space under $L^2$ norm.
So, I would appreciate any critiques I can get for my proof of the following problem.
The Problem:
Let $[a.b]$ be the closed, bounded interval of real numbers. Show that the $L^2[a,b]$ inner product ...
1
vote
1answer
37 views
Completion of inner product space
Let $(X,\langle\cdot,\cdot\rangle)$ be a real inner product space. Then, there exists a linear isometry from $X$ to its dual $X'$, whose image is dense.
I know that in the case $X$ is complete, the ...
1
vote
1answer
34 views
Subsets of complex plane with euclidean metric that are complete
Here is a problem from a past topology exam I found and was trying to tackle:
Let $A \subset S^1 \subset\mathbb{C}$ and define $X(A) := \bigcup_{z\in A} \{ tz : t \in \mathbb{R}\}$:
a) Charachterize ...
0
votes
1answer
26 views
Proof completenes of $ \{x \in \mathbb{C}^\mathbb{N}\ |\ \sum_{n=1}^\infty s_n |x_n|^p < \infty \}$
Let $(s_n)_{n\in\mathbb{N}} \subseteq \mathbb{R}$ such that for all $n$: $0 < s_n \leq \frac{1}{n} $. Let $p>1$.
How to show that the space of sequences
$ l^p_s := \{x \in \mathbb{C}^\mathbb{...
0
votes
3answers
38 views
Rudin's functional analysis appendix A4 (a)
Quick question about the following theorem:
If $K$ is a closed subset of a complete metric space $X$ then the following three properties are equivalent:
(a) $K$ is compact
(b) Every ...
0
votes
1answer
64 views
Prove that (𝑋, 𝑑) is a complete metric space [closed]
On the space of the codes $X = \{1,2,3, ..., N-1\}^n$ is defined the distance between two points $x,y \in X$ with $$d(x,y) := \sum_{i=1}^n \frac{|x_i - y_i|}{(N+1)^i}.$$
Prove that each pair $(x,y)\...
0
votes
1answer
44 views
Open Mapping Theorem fails when space is not complete
I am working on the open mapping theorem, and I want to show that completeness is a necessary condition for it to work. I have shown that if $X$ is Banach, $Y$ is a normed space, then there exists a ...
8
votes
1answer
426 views
Given $T \in L(X,Y)$, show the equivalence between: existence of $S$ such that $S(T(x))=x$, and $T$ being injective with $T(X)$ complemented in $Y$
Given $X,Y$ Banach spaces and $T \in L(X,Y)$, show that the following sentences are equivalent:
A) there exists $S \in L(Y,X)$ such that $S(T(x))=x$ for all $x \in X$.
B) $T$ is injective and $T(X)$ ...
1
vote
1answer
60 views
completeness and the open mapping theorem
Let $X,Y$ be normed vector spaces, I want to show that the open mapping theorem requires completeness of both spaces. So my question consists of two parts:
$\textit{i)}$ Let $X$ be a Banach space and ...
0
votes
1answer
35 views
Proving the statement “$l^1$ is complete.” differently.
I want to prove $l^1$ is complete. I want to prove that any Cauchy sequence $(x_n)$ is Convergent to $(y)$ whose $y_i$ is the limit of the sequence $x_{(n,i)}$ . I want to prove this part first then ...
1
vote
1answer
32 views
A complete orthonormal system $\{e_i\}^\infty_{i=1}$ in $H$ is a basis in $H$
I'm studying about Hilbert space from a book of functional analysis and I just read this theorem (2.1.10) and its' proof.
I cannot understand why $(y-x)\perp e_i$? why is it implied?
2
votes
1answer
45 views
Possible characterization of compact metric spaces via real-valued uniformly continuous functions?
1) If $X$ is a metric space such that the image of every uniformly continuous function $f: X \to \mathbb R$ is bounded, then is it necessarily true that $X$ is compact ?
2) If $X$ is a metric space ...
6
votes
2answers
58 views
On a special type of sequence in complete metric space
Let $\{x_n\}$ be a sequence in a complete metric space $X$ such that for every uniformly continuous function $f:X \to \mathbb R$, the sequence $\{f(x_n)\}$ is convergent in $\mathbb R$. Then is it ...
4
votes
2answers
356 views
Complete metric space in which the large balls are not compact.
Find an example of a complete metric space $X$ in which all sufficiently small closed balls are compact but large ones are not.
First of all, what is ''a small ball'' and ''a large ball'' in a metric ...
2
votes
1answer
70 views
Any finite dimensional normed linear space over a complete field is complete.
Prove that any finite dimensional normed linear space over a complete field is complete.
After several comments and corrections, I present the correct proof in the answer section.
3
votes
1answer
38 views
If $(X,\|.\|_*)$ is complete and $\|.\|_{*} \leq \|.\|_{**}$, then is $(X,\|.\|_{**})$ complete?
If a space $X$ is complete with respect to $\|.\|_*$ and we have that $\|f\|_{*} \leq \|f\|_{**}$ for all $f \in X$.
Does this imply that the space will be also complete with respect $\|.\|_{**}$
3
votes
1answer
43 views
On a version of Urysohn lemma for complete metric spaces, involving uniform continuous functions
Let $X$ be a complete metric space. Let us say that given a subset $A\subseteq X$, a point $a\in X$ is a limit point of $A$ if for every $r>0, A \cap B(a,r)\setminus \{a\} \ne \phi$. Now let $A,B$ ...
0
votes
1answer
19 views
Choosing a correct option related to compactness and completeness about image of map.
Let $f : X \to Y$ be a continuous map between metric spaces. Then $f(X)$
is a complete subset of $Y$ if
A. the space $X$ is compact
B. the space $Y$ is compact
C. the space $X$ is complete
D. the ...
0
votes
2answers
36 views
Are there any sets that are not complete metric spaces under all possible metrics?
I don't have any particular set in mind but this seemed interesthing since completeness depends on the metric.
0
votes
1answer
37 views
How to find a completion for a metric space (For instance, support compact continuous real functions)
A completion of a metric space $(M,d)$ is a complete metric space $(M^*,d^*)$ such that $(M,d)$ is a dense subspace of $(M^*,d^*)$. I understand this, but how do I explicitely find a completion of a ...
1
vote
1answer
29 views
Proving that the non-negative real line is complete [duplicate]
Heads up: I am very new to abstract algebra and proofs.
Take the non-negative real line $X = 0 \cup \mathbb{R}^+=[0, +\infty)$. We know that $X\subset \mathbb{R}$ and that $\mathbb{R}$ is complete. ...
3
votes
3answers
96 views
Let $Y$ be a complete metric space. Then $C^0 (X,Y)$ is complete under the uniform convergence metric.
I'm trying to show that the set of continuous functions $f: X \to Y$ is complete under the uniform convergence metric if $Y$ is complete.
Just to be clear, the metric is:
$$d(f,g) = \text{sup}\, \{d(...
-1
votes
1answer
30 views
Is a space of polynomials over Real numbers complete?
Let $P$ be the space of all polynomials over $\mathbb{R}$ normed by, $\|P\|= \max \{|a_0|,|a_1|,|a_2|,...,|a_n|\}$ where $p(x)=\sum_{k=0}^{n}a_kx^k$. Is this space complete?
Actually this problem is ...
2
votes
1answer
62 views
Is this set meager? $A = \{x\in \mathbb{R}: \exists c>0, |x-j2^{-k}|\geq c2^{-k}, \forall j\in \mathbb{Z}, k\geq 0 \}$
We define the subset $A\subset \mathbb{R}$ as follows: $x\in A \Longleftrightarrow$ if $\exists c>0$ so that
$$ |x-j2^{-k}|\geq c2^{-k} $$
holds $\forall j\in \mathbb{Z}$ and integers $k\geq 0$. ...
