Questions tagged [complete-spaces]

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

1,029 questions
Filter by
Sorted by
Tagged with
79 views

Completeness meaning (complete basis vs complete metric space)

Today my professor started talking about the formalism of QM. We talked about the eigenvectors of a Hermitian operator (over Hilbert space) as a "complete set". He also mentioned briefly ...
• 389
53 views

• 54.3k
72 views

Completeness of $\sin(kx)_{k=0}^{\infty}$ in $L_1[1, 4]$

I am reading a function analysis book and having trouble proving the following task: is ${\sin(kx)}_{k=0}^{\infty}$ complete in $L_1[1, 4]$? I understand that completeness in $C[1, 4]$ implies ...
44 views

The space $C_c$ of real-valued compactly supported, continuous functions is not a Banach space under any norm

This answer showed the space $c_{00}$ of compactly supported sequences, is not a Banach space under any norm. I wonder if the same is true for the space $C_c$ of real-valued compactly supported, ...
• 3,045
52 views

Assume that for every $f\in X^*$, there exists $y \in X$ such that $f(x)=\langle x, y \rangle$ for every $x \in X$. Show that $X$ is a complete space.

Let $(X, \langle \cdot, \cdot \rangle)$ be a real or complex vector space with an inner product. Assume that for every $f \in X^*$, there exists $y \in X$ such that $f(x) = \langle x, y \rangle$ for ...
1 vote
95 views

Rationals are incomplete and naturals are complete

Why are rationals incomplete and natural complete? I have been going through Analysis textbooks but I don't totally get the reason why. So, naturals are complete because you can divide them into two ...
• 329
1 vote
44 views

What happen if we eliminate only one point from a Cauchy complete metric space?

Consider $\mathbb{R}$ with the usual metric. We know that $\mathbb{Z}$ is complete as a subspace of $\mathbb{R}$ (because is closed), and $\mathbb{Z} \setminus \{0\}$ is still complete. But if we take ...
96 views

• 1,770
17 views

Topological tensor product

Let $A$ and $B$ be topological abelian groups. Consider the following universal property for a pair $(P, \pi)$, where $P$ is a topological abelian group and $\pi: A \times B \to P$ is a jointly ...
• 1,647
85 views

• 23
71 views

Inverse limits of complete metric spaces is Baire

It is well known that arbitrary products of complete metric spaces are Baire (refer to Dugundji, example). But, what happens when one considers inverse limits of complete metric spaces over an ...
• 444
43 views

Counterexample of Ekeland's variational principle

This is the version of the Ekeland's variational principle that i'm studying: Let $(X, d)$ be a complete metric space, and let $f: X \rightarrow \mathbb{R} \cup\{+\infty\}$ be a lower semicontinuous ...
• 641
34 views

Completeness preserved under specific homeomorphism

Problem: Let $(X,d)$ be a complete metric space, $(Y,d')$ a metric space, and $f\colon X\to Y$ a homeomorphism, such that there exists a $c>0$, for which $$c\cdot d(x,y)\leq d'(f(x), f(y)).$$ Show ...
44 views

Find an example to show Cantor intersection theorem does not hold if we eliminate one condition

Cantor's intersection theorem states that: Let $(X,d)$ be a complete metric space, and let $A_1\supset A_2\supset...$ be an infinite chain of nonempty, closed, bounded subsets of $X$. Suppose further ...
77 views

• 303
1 vote
120 views

Is the Axiom of Completeness logically equivalent to "There is no proper superset of $\mathbb R$ that is an ordered Archimedean field"?

The Axiom of Completeness can be formulated as: There exists a set $R$ such that: $R$ is an ordered Archimedean field Any nonempty subset of $R$ with an upper bound has a least upper bound. Recently,...
• 4,450
50 views

Subspace of real valued functions on [0,1] is complete

I’m currently working on the following problem: Let $X$ denote the set of nondecreasing functions $f:[0,1]\to\mathbb{R}$. We endow $X$ with the sup metric. Prove that $X$ is complete. I notice that if ...
154 views

Can we include a metric space into its completion?

Let $(X,d)$ be a metric space and let $(X^*,d^*)$ denote its completion (via equivalence classes of Cauchy sequences in $X$). Let $f:X \to X^*$ be an isometry such that $f(X)$ is dense in $X^*$. Now ...
• 4,827
46 views

Compact-open topology not always complete with Arens' metric

For an infinite-type surface $S$ with complete metric $d$ and compact exhaustion $\{K_n\}_{n=1}^\infty$, Vlamis, in "Notes on the Topology of Mapping Class Groups" Appendix A, defines a ...
• 1
85 views

Using only the universal property, prove that $X$ is dense in its Cauchy completion

Long ago, I learned about the Cauchy completion of metric spaces via the usual explicit construction of quotienting the set of Cauchy sequences. For a metric space $X$, let $\hat X$ denote this Cauchy ...
• 4,057
1 vote
38 views

Applying Hilbert's geometric Axiom of Completeness to deduce equivalent forms

In Hilbert's Foundations of Geometry, the axiom of completeness is presented as follows: To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner ...
92 views

On the completeness of a metric space.

Theorem: Consider the metric space $(X, d)$, where $X \subseteq \mathbb{R}$ and $d(x, y) = |f(x) - f(y)|$, with $f$ being a one-to-one (injective) function on its domain $X$. If the range of $f$ is a ...
• 6,088
1 vote
121 views

Completeness of $(C^{\infty}(\mathbb{R}),d)$

Let $C^{\infty}(\mathbb{R})$ be the space of all infinitely differentiable complex-valued functions on $\mathbb{R}$. Define the metric \begin{align} d(f,g) = \sum_{n,m=0}^{\infty} 2^{-n-m}\frac{\sup_{...
• 907
1 vote
59 views

Find the completion of a metric space [closed]

Prove that $\mathbb{R}$ with metric $\rho(x,y)=|\arctan x-\arctan y|$ is not a complete metric space. Determine its completion. Here is the link to prove the given metric is not complete. Here is the ...
1 vote
45 views

66 views

Countable family of norms and seminorms

In volume 4 of Gelfand and Vilenkin they study countably normed spaces. I assume a definition for a countably normed space was given in an earlier volume, but I do not have access to them and I could ...
• 6,255
64 views

We say a metric space X is completely metrizable if there a metric that induces its usual topology and X with that metric is complete. I´ve seen, for example, that (0,1) is completely metrizable and ...
• 69
1 vote
96 views

Discontinuity points of a function over dense topologically complete subset.

I spend a lot time with this problem, and I need your help, Let $(X,d)$ be a metric space and $A$ be a dense subset of $X$, suppose that there is a bounded distance $d’$ on $A$ topologically ...
85 views

Find a Closed and Bounded set but not Compact in an incomplete metric space

I am learning topology by myself: Let $(X,d)$ be an incomplete metric space. Show that there exists a closed and bounded set $E \subset X$ that is not compact. My attempt: Since $X$ is incomplete ...
• 43
73 views

1 vote
57 views

If Sobolev space is isometric to $L^2(\mathbb{R})$ then Sobolev space is complete?

My attempt: Let $H^2(\mathbb{R})=\left\{u\in L^2(\mathbb{R}): \left\|u\right\|_{H^2(\mathbb{R})}:=\left\|\mathcal{F}^{-1}((1+\xi^2)\widehat{u})\right\|_{L^2(\mathbb{R})}<\infty\right\}$ the Sobolev ...
• 3,630
1 vote
36 views

Are time-dependent Banach space-valued functions a Frechét space?

Consider a family of Banach spaces $(\mathcal{B}_{t},\Vert\cdot\Vert_{t})_{t\in\mathbb{R}}$ and define the space $C^{\infty}(\mathbb{R},\mathcal{B}_{\bullet})$ where smootheness has to be understood ...
• 2,876
1 vote
Consider the sequence of functions $<f_n(t)>$, defined as \$ f_n(t) = \begin{cases} e^{-t^2} & \text{if } -n \leq t \leq n \\ \frac{e^{-n^2}}{[1-n(t-n)]} & \text{if } n \leq t &...