Questions tagged [complete-spaces]
A metric space is complete if, in it, any Cauchy sequence is convergent.
917
questions
1
vote
1
answer
30
views
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\...
- 11
-2
votes
0
answers
37
views
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 ,...
- 11
0
votes
0
answers
23
views
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
$$
||...
- 645
1
vote
3
answers
46
views
Prob. 14, Sec. 1.5 in Erwine Kreyszig's Introductory Functional Analysis With Applications
Let $C([0,1])$ be the set of all continuous real-valued function on $[0,1]$, let $d(f,g)=\int_0^1 |f(t) - g(t)|dx$.
We note that the exercise 14 needs exercise 13, which gives us a sequence $(f_n)_{n\...
- 100
4
votes
1
answer
59
views
Showing that a sequence is Cauchy on $C[0,1]$.
Let $$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}$$We will prove the sequence $(f_k)_{k \in \mathbb{N}}$ is Cauchy on $(C[0,1],\|\cdot\|_p)$ ...
- 408
4
votes
0
answers
63
views
(Cauchy) complete metric spaces in enriched category theory
A Lawvere metric space is Cauchy complete iff it is Cauchy complete regarded as a $[0,\infty]$-enriched category. I have two questions about this observation:
Is Cauchy completeness (in the setting ...
- 152k
2
votes
1
answer
47
views
Verification of the converse of Baire Category Theorem
One way to state the Baire Category Theorem is as follows: If $X \neq \emptyset$ is a complete metric space, then $X$ is nonmeager as a subset of itself.
Then it was mentioned that the converse, that ...
- 408
0
votes
0
answers
15
views
equivalent metrics and complete metrics. [duplicate]
$X\neq\emptyset$. Let $d_1$ and $d_2 $ be equivalent metrics on $X$. $(X$, $d_1)$ in complete $\Longleftrightarrow$ $(X$, $d_2)$ in complete.
- 29
2
votes
1
answer
29
views
Deciding if $l^1(\mathbb{Z}_+)$ over $\mathbb{F}$ is complete with the norm $\lVert x\rVert=2\left|\sum_{n\geq 1}x_n\right|+\sum_{n\geq2}(1+1/n)|x_n|$
Consider the space $l^1(\mathbb{Z}_+)$ over a field $\mathbb{F}$ (real or complex) and take it as granted that the function
$$\lVert x\rVert=2\left|\sum_{n\geq 1}x_n\right|+\sum_{n\geq 2}(1+1/n)|x_n|$$...
- 701
1
vote
1
answer
57
views
Is $C([0,1])$ complete if $\int_{0}^{1} f(x)g(x)dx$ is the inner product?
One of my homework questions asked me to show that $$(f, g) \mapsto \int_{0}^{1} f(x)g(x)dx$$
defines an inner product on $C([0,1])$, which I was able to do successfully. However, I am curious on ...
- 1,296
0
votes
2
answers
32
views
Prove the following space is not complete.
I am asked to prove the following space of sequences is not complete:
$$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}\}$$
I followed ...
- 1,460
0
votes
0
answers
25
views
Why is it necessary for the domain to be bounded for this banach space?
I have an elementary question regarding this definition, from the site of the University of Bath, UK:
Theorem 2.18. $C^k(\Omega)$ is Banach. Let $\Omega \subset \mathbb{R}^N$ be a bounded open set and ...
- 138
1
vote
1
answer
35
views
Uniform continuity and seminorms
I recall that a seminorm is basically a norm that is not necessarily positive definite.
Let $(E,(p_n)_{n\in\mathbb{N}})$ be a Fréchet space, meaning each $p_n$ is a seminorm and if we equip $E$ with ...
- 133
2
votes
1
answer
36
views
Weak quasi-completion of a locally convex space
Let $X$ be a locally convex space.
Then as far as I understand the bidual of $X$ with the weak topology, $(X_\beta')_\sigma'$, is like a quasi-completion of $X_\sigma$.
Namely, if $B \subseteq X$ is ...
- 173
2
votes
1
answer
105
views
Is this inner product space also a Hilbert space?
Let $X$ be the set of all the real- (or complex-) valued functions that are defined and continuous on a closed interval $[a, b]$ on the real line, where $a$ and $b$ are some fixed real numbers such ...
- 25k
0
votes
0
answers
28
views
Are numbers only noncomputable in context?
I’m not sure how to word this question, but here goes.
Let’s say I’m a programmer who wants to use a Turing machine to compute the probability $\Omega$ that some Lisp program will halt. Obviously, I ...
0
votes
1
answer
39
views
Find the completion of the metric spaces
Metric spaces are:
a) Let $X = C[0,1]$ and $$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
112
views
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 $\...
- 307
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 $A^{n}=\emptyset?$ ...
- 39
0
votes
1
answer
25
views
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 \...
- 77
0
votes
0
answers
52
views
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 $\...
- 379
4
votes
1
answer
120
views
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 ...
- 47.3k
0
votes
2
answers
38
views
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,...
- 1,074
0
votes
1
answer
30
views
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_{[...
- 5,508
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.
...
- 1,767
0
votes
1
answer
58
views
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 ...
- 5,888
1
vote
0
answers
39
views
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 ...
- 3,520
0
votes
1
answer
40
views
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 ...
- 8,562
0
votes
0
answers
55
views
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....
- 4,230
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_{\...
- 73.3k
1
vote
2
answers
68
views
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 ...
- 4,230
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.
...
- 101
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
82
views
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 \...
- 5,090
0
votes
0
answers
34
views
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 ...
- 415
1
vote
1
answer
39
views
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}$ ...
- 3,520
1
vote
1
answer
43
views
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 ...
- 2,297
0
votes
0
answers
46
views
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 \...
- 79
1
vote
1
answer
59
views
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 (...
- 39
1
vote
1
answer
45
views
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 ...
user1054388
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=(\...
- 125
0
votes
1
answer
45
views
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[
...
- 45
0
votes
1
answer
38
views
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\...
- 664
0
votes
0
answers
42
views
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 ...
- 613
1
vote
0
answers
50
views
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 ...
- 11
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 $W$ ...
- 472
0
votes
0
answers
30
views
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 $\left\{ E_i \right\}$...
- 5,090
2
votes
1
answer
88
views
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 ...
- 158