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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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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) \...
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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(\...
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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(...
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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 $c_0$ (...
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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 ...
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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 ...
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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}(...
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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 ...
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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\...
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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 ...
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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 ...
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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 $...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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''}), \; \...
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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. $\mathbb{R}^{d}$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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},\...
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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^...
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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 ...
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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 ...
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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 ...
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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{...
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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 ...
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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)\...
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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 ...
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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 $T(X)$ ...
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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 ...
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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 ...
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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?
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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 ...
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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 ...
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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
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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.
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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 $\|.\|_{**}$
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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 $A,B$ ...
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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 ...
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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.
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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 ...
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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(...
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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 $k\geq 0$. ...