# Questions tagged [complete-spaces]

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

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### Why complete space is good?

I wonder why the complete space is a good space. I know that for a metric space $X$, if any Cauchy sequence $\{x_n\}$ in $X$ converges, then $X$ is called complete metric space. But why such property ...
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### Completeness of Metric Space

Consider the set $C^k(\mathbb{R^n},\mathbb{R^m})$ in the metric $\rho$ where $\rho$ is defined by $$\rho(f,g)=\sup\{d(f(x),g(x)),d(f^1(x),g^1(x)),\dots ,d(f^k(x),g^k(x))|x\in \mathbb{R^n}\}.$$ How can ...
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### Completion of topological spaces.

Let be $Y \subset X$ be dense between rings with I-adic topology or between metric spaces. Can I conclude that the completion of $X$ is the same as the completion of $Y?$ I would think that it seems ...
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### Homogeneous Besov space $\dot B^{-\sigma}(\mathbb R^d)$ is a Banach space

Let $\sigma > 0$ and $\theta \in S(\mathbb R^d)$ (Schwartz function) be fixed, where $\theta$ is such that its Fourier transform $\hat{\theta}$ is compactly supported, $0 \leq \hat{\theta} \leq 1$ ...
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### Proving a metric space is a complete metric space.

Let $X$ be a compact metric space, and let $B(X)$ be the set of real-valued bounded functions on $X$. For any $f, g ∈ B(X)$, define $$d_B(f, g) :=\sup _{x\in X}\left | f(x)-g(x) \right |$$ Suppose, we ...
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### Is it true that a metric over $\mathbb{R}^n$ (or $\mathbb{C}^n$) always makes a complete metric space?

I'm studying for my calculus II exam and I was thinking about the statement in the title. My "sloppy" proof is something like: Over $\mathbb{R}^n$ (or $\mathbb{C}^n$) the Bolzano-Weierstrass ...
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### Let $E = \{(x,y) \in \mathbb{R}^2 \mid y=x^2 \}$. Show that $(E,d_E)$ is a complete metric space.

Let $E = \{(x,y) \in \mathbb{R}^2 \mid y=x^2 \}$. Show that $(E,d_E)$ is a complete metric space. Here $d_E$ is just the restriction of $d_2$ to $E$. I'm trying to show this using the preimage ...
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### Is there a norm in which the vector space of all sequences with the induced metric is complete?

The question of knowing whether is possible put a norm in a vector space is something new to me. I liked the Ivo ideas and I have wondered a little more about a particular case. Is there a norm in ...
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### Why is the unitization of $M_\infty(\mathbb{C})$ not complete?

Consider the space $M_\infty(\mathbb{C})$ of infinite matrices with only finitely many nonzero entries. This is a pre-C$^*$-algebra, and we can consider its unitization $M_\infty^1(\mathbb{C})$ (the ...
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### Relationship between the notions of completeness for a metric space, a vector space and a field

My question is motivated by the varying notions of 'completeness' one attaches to these objects. Cauchy completeness: Pertaining to metric spaces. R with the Euclidean metric is Cauchy complete. LUB-...
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### $X$ is complete iff $\sum_{n=1}^\infty \|x_n\| < \infty \implies \sum_{n=1}^\infty x_n$ converges (Carothers, Theorem $7.12$)

I have some questions to ask about the second part of the proof, i.e. $[\Leftarrow]$ direction: The author says, "As always, it is enough to find a subsequence of $(x_n)$ that converges". ...
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### if two complete metric spaces coincide as sets, are their metrics (strongly) equivalent?

If two complete metric spaces coincide as sets, are their metrics (strongly) equivalent? If $p$ and $q$ are two strongly equivalent metrics (in the sense that $A p(x,y) \leq q(x,y) \leq B p(x,y)$ for ...
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### I want to prove completeness of $C[a,b]$ with metric induced by max norm, but need help

I added a picture, please check it. Can someone please explain the last three line of the proof? I thought by finding a cauchy sequence that converges to a continuous function we are done with proving ...
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### Why we need complete space for the interchange of limits to be valid?

I am learning Real analysis using the book Analysis 2 by Terence Tao. I'm confused with the condition on which we can interchange the limit. Concerning the below proposition, my question is that: Why ...
Let: $$D_t = \{(x,y,t) \in \mathbb{R}^3 : x^2 +y^2 \leq 1\},$$ $$S=\{(p,q,0)\in \mathbb{R}^3: p^2 +q^2<1, p\in \mathbb{Q}, q\in \mathbb{Q} \}$$ $$I(a,b) \text{ - open line segment between points } ... 1answer 57 views ### The set C= \{x \in X: \text{dist} (x,K) ≤\frac 12 \} is not compact Give an example of a complete metric space (X,d) and its compact subset K, that the set C= \{x \in X: \text{dist} (x,K) ≤\frac 12 \} is not compact. From Heine-Borel theorem: completeness + ... 1answer 17 views ### Check the following statements. Let X=\{(x_1,x_2,\dots):x_i\in\Bbb{R} and finitely many x_i's are non zero} and d:X\times X\to\Bbb{R} be a metric on X defined by d(x,y)=\text{sup}_{i\in\Bbb{N}}|x_i-x_j|,x=(x_1,x_2,\dots),y=(... 1answer 57 views ### Prove that (X,d) is complete Let (X,d) - metric space such that A,B \subset X and  X=A\cup B where (A,d_{|A\times A}), (B,d_{|B\times B}) are complete subspaces. Prove that (X,d) is complete. I think it should be ... 1answer 36 views ### Banach contraction theorem with composition Problem 5. Let (\mathcal{X},\mathrm{d}_\mathcal{X}) be a non-empty complete metric space. Suppose that f,g:\mathcal{X}\longrightarrow\mathcal{X} are two Banach's contractions of \mathcal{X}. ... 3answers 54 views ### \mathbb{Q}[\sqrt{2}] is not a complete space I am trying to prove that \mathbb{Q}[\sqrt{2}]=\{a+b\sqrt{2}:a,b\in\mathbb{Q}\} is not a complete space (with the standard metric). For that purpose, I am looking for a Cauchy sequence in \mathbb{Q}... 0answers 20 views ### Is Space (X, \bar{\rho}) complete? ( What limit of sequence should I take?) Let X = {x\in \mathbb{R}^{\omega} , there exists N \in \mathbb{N} such that x_i=0 for all i\geq N}, and let \bar {\rho} be the uniform mertic on  \mathbb{R}^{\omega} . Is (X, \bar{\rho})... 1answer 62 views ### Lebesgue outer measure and complete metric space Let A be an elementary set. For E,F\subseteq A, let E\sim F iff E\triangle F is null. Define d([E]_{\sim},[E']_{\sim})=m^*(E\triangle E'). I've shown that ((\mathcal PA)/\sim,d) is a ... 1answer 56 views ### Space of Katetov-Functions with sup-metric is complete metric space Let (X,d) be a metric space. f: X\longrightarrow\mathbb{R} is called "Katetov map" iff :$$∀𝑥,𝑦,∈𝑋:|𝑓(𝑥)−𝑓(𝑦)|≤𝑑(𝑥,𝑦)≤𝑓(𝑥)+𝑓(𝑦) The set of all Katetov-maps on X is denoted ...
This is a short one but: Consider the real matrix $\alpha = \begin{pmatrix} 7 &3 &-4 \\ -2&-1 &2 \\ 6&2 &-3 \end{pmatrix}$ and let $\beta \in M_n (\mathbb{C})$ be the ...