Questions tagged [complete-spaces]

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

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is the union of two totally bounded sets totally bounded?

i am trying to prove that the union of A and B is totally bounded when A and B are both totally bounded. Firstly , A is totally bounded so there is a e-dense subset so that A subset of`$U_{i=1}^{n}B\...
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Is $(\{1/n^2 : n \in \mathbb N\},|\cdot|)$ a complete metric space?

I am trying to check whether this metric space $(\{1/n^2 : n \in \mathbb N\},|\cdot|)$ is complete or not. Do I need to prove that an cauchy converges? i am thinking to set A={1/n^2} as a susbet of R ,...
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On the completeness of a $L^2(\mathbb{R}^3)$ subspace

Consider the space $$ S(\mathbb{R}^3) = \{u \in L^2(\mathbb{R}^3) : u_{x_1}, u_{x_2} \in L^2(\mathbb{R}^3), |(x_1,x_2)|^{2} u_{x_3} \in L^2(\mathbb{R}^3)\}. $$ Is this space we consider the norm $$ ||...
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3 answers
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Prob. 14, Sec. 1.5 in Erwine Kreyszig's Introductory Functional Analysis With Applications

Let $C([0,1])$ be the set of all continuous real-valued function on $[0,1]$, let $d(f,g)=\int_0^1 |f(t) - g(t)|dx$. We note that the exercise 14 needs exercise 13, which gives us a sequence $(f_n)_{n\...
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1 answer
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Showing that a sequence is Cauchy on $C[0,1]$.

Let $$f_k(t) = \begin{cases}0, \text{ if } t \in [0,1/2]\\ 1, \text{ if } t \in [1/2 + 1/k, 1].\end{cases}$$We will prove the sequence $(f_k)_{k \in \mathbb{N}}$ is Cauchy on $(C[0,1],\|\cdot\|_p)$ ...
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(Cauchy) complete metric spaces in enriched category theory

A Lawvere metric space is Cauchy complete iff it is Cauchy complete regarded as a $[0,\infty]$-enriched category. I have two questions about this observation: Is Cauchy completeness (in the setting ...
2 votes
1 answer
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Verification of the converse of Baire Category Theorem

One way to state the Baire Category Theorem is as follows: If $X \neq \emptyset$ is a complete metric space, then $X$ is nonmeager as a subset of itself. Then it was mentioned that the converse, that ...
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equivalent metrics and complete metrics. [duplicate]

$X\neq\emptyset$. Let $d_1$ and $d_2 $ be equivalent metrics on $X$. $(X$, $d_1)$ in complete $\Longleftrightarrow$ $(X$, $d_2)$ in complete.
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1 answer
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Deciding if $l^1(\mathbb{Z}_+)$ over $\mathbb{F}$ is complete with the norm $\lVert x\rVert=2\left|\sum_{n\geq 1}x_n\right|+\sum_{n\geq2}(1+1/n)|x_n|$

Consider the space $l^1(\mathbb{Z}_+)$ over a field $\mathbb{F}$ (real or complex) and take it as granted that the function $$\lVert x\rVert=2\left|\sum_{n\geq 1}x_n\right|+\sum_{n\geq 2}(1+1/n)|x_n|$$...
1 vote
1 answer
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Is $C([0,1])$ complete if $\int_{0}^{1} f(x)g(x)dx$ is the inner product?

One of my homework questions asked me to show that $$(f, g) \mapsto \int_{0}^{1} f(x)g(x)dx$$ defines an inner product on $C([0,1])$, which I was able to do successfully. However, I am curious on ...
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2 answers
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Prove the following space is not complete.

I am asked to prove the following space of sequences is not complete: $$D = \{ a = \{a_n\}_{n\in\mathbb{N}};\ a_n\in\mathbb{C};\ a_n\neq 0\quad \text{for a limited number of elements}\}$$ I followed ...
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Why is it necessary for the domain to be bounded for this banach space?

I have an elementary question regarding this definition, from the site of the University of Bath, UK: Theorem 2.18. $C^k(\Omega)$ is Banach. Let $\Omega \subset \mathbb{R}^N$ be a bounded open set and ...
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1 vote
1 answer
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Uniform continuity and seminorms

I recall that a seminorm is basically a norm that is not necessarily positive definite. Let $(E,(p_n)_{n\in\mathbb{N}})$ be a Fréchet space, meaning each $p_n$ is a seminorm and if we equip $E$ with ...
2 votes
1 answer
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Weak quasi-completion of a locally convex space

Let $X$ be a locally convex space. Then as far as I understand the bidual of $X$ with the weak topology, $(X_\beta')_\sigma'$, is like a quasi-completion of $X_\sigma$. Namely, if $B \subseteq X$ is ...
2 votes
1 answer
105 views

Is this inner product space also a Hilbert space?

Let $X$ be the set of all the real- (or complex-) valued functions that are defined and continuous on a closed interval $[a, b]$ on the real line, where $a$ and $b$ are some fixed real numbers such ...
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Are numbers only noncomputable in context?

I’m not sure how to word this question, but here goes. Let’s say I’m a programmer who wants to use a Turing machine to compute the probability $\Omega$ that some Lisp program will halt. Obviously, I ...
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1 answer
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Find the completion of the metric spaces

Metric spaces are: a) Let $X = C[0,1]$ and $$d(x,y) = \sup_{0\le t \le1} t \vert x(t) - y(t)\vert$$ b) Let $$X = \{x \in C[0,1]: \sup_{0\le t \le1}t^{-2}\vert x(t)\vert < \infty\}, d(x,y) = \sup_{0\...
5 votes
1 answer
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Find a completion of the following spaces

Find a completion of the incomplete metric space $(X,d)$, where a) $X=\Bbb{Q}$, where $\forall{q,r}\in{X}$ $d(q,r) = \lvert {\arctan(q) - \arctan(r)}\rvert$ b) $X=(0,1)\setminus \mathbb{Q}$, where $\...
2 votes
1 answer
76 views

limit points of a countable set

Let $X$ be a complete space and $A\subset X$ be a closed countable set. Let $A^{'}$ denote the set of all limit points of $A.$ Prove that there exists an integer $n$ such that $A^{n}=\emptyset?$ ...
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Completion of the set of continous functions on $[0,1]$ with the metric $d(x(t),y(t)) = \sup_{t \in [0,1]} t^2|x(t)-y(t)|$.

I am trying to find the completion of $C[0,1]$ with the metric $d(x(t),y(t)) = \sup_{t \in [0,1]} t^2|x(t)-y(t)|$. I understand that this metric space is not complete because the function $f(x) = 1 \...
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Show that the Poincare disk is complete metric space with respect to the hyperbolic metric.

I want to show that the unit disk equipped with the hyperbolic metric is a complete metric space. The metric is defined as: $d_{P}(w,z)=inf\{l_{h}(\gamma): \gamma$ is a partial smooth curve in $\...
4 votes
1 answer
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Does every incomplete real inner product space admit a closed subspace of countable dimension?

Suppose $X$ is an incomplete real inner product space. Does there exist a sequence $(y_n)_n \in X$ such that $Y = \operatorname{span}\{y_n\}_n$ is closed? This question cropped up as part of my ...
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2 answers
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Find the Completion of (X, d) where $X = \{\{x_n\} : \sum_{n=1}^\infty 2^n \vert{x_n}\vert < \infty\}$ and $d(x,y) = sup_n \vert x_n - y_n \vert$ [closed]

I have showed that sequence such that $x^{(k)}_n = (\frac{1}{2 + \frac{1}{k}})^n$ is Cauchy but it converges to $ \frac{1}{2^n}$ which is not in X. But I was not able to find its completion. How can I ...
1 vote
2 answers
56 views

Question regarding one approach to showing that the vector space $C[0, 1]$ is not complete w.r.t. the $L1$ integral norm

First to clarify few things: I know that this question has been asked several times at this forum, take for example What is the completion of $C[0,1]$ equipped with the integral norm? The space $C[a,...
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1 answer
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Completeness of normed linear space $C^1[0,1]$

The Space $C^1[0,1]$ of continuously differentiable function is complete with respect to norm ? $1.$ $\|f\|=sup_{[0,1]}|f’(x)|$. $2.$ $\|f\|=sup_{[0,1]}|f(x)|$. $3.$ $\|f\|=sup_{[0,1]}|f(x)|+sup_{[...
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2 votes
0 answers
46 views

Completeness of French railroad metric space

The French railroad metric on $\mathbb{R}^2$ is defined as $||x - y||$ is $x, y$ are on a line that goes through the origin, or otherwise as $||x|| + ||y||$. Prove this forms a complete metric space. ...
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0 votes
1 answer
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Why is $K^{\mathrm{sep}}$ dense in $K_v^{\mathrm{sep}}$?

Let $K$ be a global field and $v$ be a non-archimedean place of $K$. Given a field $k$, let $k^{\mathrm{sep}}$ be a separable closure inside an algebraic closure $\overline k$. I would like to know ...
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1 vote
0 answers
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How the weak$^*$ topology of the space of all Borel probability measures on $X$ is defined?

Let $(X, d)$ be a metric space. Then @Nate Eldredge said that The space $\mathcal{P}(X)$ of probability measures on a Polish space $X$, endowed with the weak topology (induced by bounded continuous ...
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1 answer
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Normalization of Vector in arbitrary Normed Vector Space

Let $K$ be a field equipped with an absolute value $\vert-\vert:K\rightarrow \mathbb{R}_{\geq 0}$ satisfying $\vert k \vert = 0$ iff $k=0$ $\vert x+y \vert\leq\vert x\vert+\vert y\vert$ $\vert x\cdot ...
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0 answers
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Prove that any normed vector space with finite dimension is complete.

THIS QUESTION IS NOT A DUPLICATE OF THIS ONE: INDEED, I PROPOSED TO ANALYSE AN ALTERNATIVE PROOF AND IN PARTICUAR I ASK FOR A TOPOLOGICAL PROOF WHICH SUBSTANTIALLY USE NOT MANY ALGEBRAIC MANIPULATIONS....
13 votes
1 answer
236 views

Another completion of $\overline{\Bbb{Q}}$

$\overline{\Bbb{Q}}$ is the field of algebraic numbers, let $E$ be the set of embeddings $\overline{\Bbb{Q}}\to \Bbb{C}$ and consider the following norm on $\overline{\Bbb{Q}}$ $$\|\alpha\|=\sup_{\...
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1 vote
2 answers
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If $d_1$ and $d_2$ are two equivalent metric on $X$ then is $X$ complete with respect $d_1$ if and only if it is complete with respect $d_2$?

I know that completeness is not a topological property so that if $d_1$ and $d_2$ are two equivalent metric of $X$ then I can state that $X$ is complete with respect $d_1$ if and only if it is ...
1 vote
1 answer
60 views

Prove that $\mathbb{Q}$ with the topology induced by the usual distance is not complete.

My question is not really about proving this fact. Instead, it is about the sense of the sentence. What does "the topology induced by the usual distance" mean? I never understood it clearly. ...
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2 votes
1 answer
55 views

Is it true, that, metrizable field is locally compact iff complete (in topology Induced by Metric)?

It's shown in cource of topology that metric space is compact iff it satisfies Finite Intersection Axiom iff it is Sequentially Compact. In cource of calculus it's correspondingly shown that in ...
1 vote
2 answers
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Use of theorem 3.5 Rudin's Functional Analysis to show simple functions are dense in $L^p$

Can anyone give me a hint for what I thought was an application of the following theorem Theorem 3.5 (Rudin's Functional Analysis): Suppose $M$ is a subspace of a locally convex space $X$, and $x_0 \...
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Showing that the metric $d(n, m) = 2^{-r}$ if $|n - m| = 2^{r}t$ with $t$ odd if $n \neq m$ and $d_2(n, n) = 0$ on $\mathbb Z$ is not complete [duplicate]

I'm looking at some old question papers and here's a problem: If $n$ and $m$ are integers and $d(n, m) = 2^{-r}$ if $|n - m| = 2^{r}t$ with $t$ odd if $n \neq m$ and $d_2(n, n) = 0$. Prove that the ...
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1 vote
1 answer
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Where completeness and separability are used in a proof of this lemma?

I'm reading Lemma 2.30. from this note. Lemma 2.30. Let $\mathcal{V}$ and $\mathcal{D}$ be two algebras of subsets of a separable complete metric space $T$. Assume $\mathcal{V} \subseteq \mathcal{D}$ ...
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1 vote
1 answer
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Fast ways to know if a space is complete?

My professor listed six normed function spaces on the board and asked, "Which are complete? For those that are not complete, what is their completion?" I'm wondering if there is a fast way ...
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Proof that this is a complete metric space

How do I prove that a metric space defined by ($A,d$), where $A$ is any set and $d$ is the distance defined by $\displaystyle d(x,y) = \left\{ 1, \ \text{if} \ x \neq y \atop 0, \ \text{if} \ x = y \...
1 vote
1 answer
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How to show that the cartesian product of Hilbert spaces is a Hilbert space?

Let $E_1,...,E_n$ be inner product spaces, $\langle [x_1,...,x_n],[y_1,...,y_n]\rangle$ defines an inner product in $E = E_1 \times ... \times E_n$. Show that if $E_1,...,E_n$ are Hilbert spaces (...
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1 vote
1 answer
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Existence of an open set G such that a continuous function $f:X\to\,\mathbb{Q}$ is constant on G.

Let $(X,d)$ a complete metric space. We have a continuous function $f:X\to\,\mathbb{Q}$ , the rationals. Show that there exists an open set $G$ in $X$ such that $f$ is constant on $G$. My proof relies ...
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0 votes
1 answer
31 views

Complex sequence space is complete (with a certain metric)

Problem 12, section 1.5 from Introductory Functional Analysis - Kreyszig. The problem: Let $s:= S(\mathbb{C})$ be the space of all complex sequences, with metric $d$ defined for each $x=(\xi_j)_j, y=(\...
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Showing a space of functions is a Hilbert space

Let $R$ be a self-adjoint nonnegative definite ($n\times n$) matrix. Consider the class of $n\times 1 $ functions $u(\cdot)$, Lebesgue measurable on $(0,1)$, and such that $$ \| u \|^2 = \int_{0}^1[ ...
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1 answer
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Excercise 15 Rudin functional analysis chapter 2

I am self-studying the book function analysis of Rudin. I got stuck on the final passage of the following exercise. Suppose $X$ is an $F-$space (a topological vector space with a topology induced by a ...
1 vote
1 answer
55 views

Non-equivalent norms on $\mathbf{Q}[\sqrt2]$

I found the following post on MO, Can somebody help me filling the details? Just to be sure, here is the restated problem We work on $\mathbf{Q}[\sqrt2]=\left\{x\in\mathbf{R}\mid\: \exists (a,b)\in\...
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0 answers
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Verifying a function space to be a Hilbert space

Let $\omega(t)$ be a non-negative real Lebesgue-integrable function. $$H=\{f,\ \textrm{Lebesgue-measurable}: \int_{\mathbb{R}}|f(t)|^2\omega(t)dt<\infty\}$$ Prove that $H$ is a Hilbert space under ...
1 vote
0 answers
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A metric space $F$ is complete if and only if it is closed w.r.t any bigger space $X$.

Let $(F,d_F)$ be any metric space. Then $F$ is complete if and only if, for any space $(X,d_X)$ with $F\subset X$ and ${d_{X_{|F\times F}}}=d_F$, $F$ is closed in $X$. $"\Rightarrow"$ Let $(F,d_F)$ be ...
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2 votes
1 answer
59 views

Completion of the space of bounded linear operators by completing the image.

Let $V, W$ be normed vector spaces and $B(V,W)$ be the space of bounded linear operators from $V$ to $W$. In particular, $W$ may be a Banach space or not, i.e. not complete. I have learned that if $W$ ...
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Is the space of complex measures on a sigma algebra complete?

Let $\mathcal{M}$ a $\sigma$-algebra on a set $X$. A complex measure is a complex function on $\mathcal{M}$ such that $$ \mu(E) = \sum_i \mu(E_i) $$ where $E \in \mathcal{M}$ and $\left\{ E_i \right\}$...
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2 votes
1 answer
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How to understand the completion of normed space

In the book I'm reading, the theorem is as follow: If $\mathcal{X}$ is a normed space, then there is a Banach space $\hat{\mathcal{X}}$ and a linear isometry $U:\mathcal{X}\to \hat{\mathcal{X}}$ such ...
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