# Questions tagged [complete-spaces]

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

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### Showing incompleteness of metric space by specifying non-convergent Cauchy sequence

I have been looking at the space of continuous functions over a compact interval $C([0,2])$ equiped with the the integral norm of absolut values $\| \cdot \|_1$. I read a counterexample that showed ...
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### Let $(Y,d|_{Y\times Y})$ be a subspace of $(X,d)$. If $(Y, d|_{Y\times Y})$ is complete, then $Y$ must be closed in $X$.

(a) Let $(X,d)$ be a metric space, and let $(Y,d|_{Y\times Y})$ be a subspace of $(X,d)$. If $(Y, d|_{Y\times Y})$ is complete, then $Y$ must be closed in $X$. (b) Suppose that $(X,d)$ is a complete ...
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### $\sup L^1$ space with uniform integrability

I have a following question. For $t\in[0,T]$, let $f_t:\mathbb{R}\rightarrow\mathbb{R}$ be uniformly integrable family i.e. $\{f_t,\,t\in[0,T]\}$- uniformly integrable. We consider the space of such ...
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### Completeness of Metric in $C^1$

Consider $C^1[0,1]$ with the metric $d(f,g)=d_{\infty}(f,g)+d_{\infty}(f',g')$. Then $C^1$ is complete. My attempt: Let $(f_n)_n$ be cauchy in $C^1[0,1]$ wrt $d$. Then $f_n$ and $f'_n$ are cauchy ...
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### Is the metric space of polynomial of degree $\leq1$ complete with the following metric?

I must determine if the following metric space is complete $$\mathbb R_1[x]\times \mathbb R_1[x]\rightarrow \mathbb R: d(p(x),q(x))=max\{|p(0)-q(0)|,|p(1)-q(1)|\}$$ where \mathbb R_1[x]=\{p(...
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### completeness of the metric $[1,2]\times[1,2]\rightarrow \mathbb R:$ $d(x,y)=|x^4-y^4|$

I must determine if the following space is complete $[1,2]\times[1,2]\rightarrow \mathbb R:$ $d(x,y)=|x^4-y^4|$. Is $([1,2],d)$complete? my try: Let $\{x_n\}$ be a Cauchy sequence by ...
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### What is wrong with my solution about the completeness of this metric space?

For $x_1,x_2 \in X = (1, 3) \subset \mathbb R$ it is defined a metric $d(x_1, x_2) = |\frac{1}{x_1}-\frac{1}{x_2}|$ I must determine if $(X; d)$ is a complete metric space The solution I was ...
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### Tell if a set $K \subseteq\mathbb R$, closed in $\tau_d$ and bounded by the distance d is necessarily compact in $\tau_d$

Considering the following metric over $\mathbb R$: $d(x,y)=|x-y|/(1+|x-y|)$ I have to 1) find if ($\mathbb R$,d) is a complete metric space 2)Tell if a set $K \subseteq\mathbb R$, closed in ...
### Completeness of $L^2([0,1]) \otimes L^2([0,1])$.
I'm trying to understand the Hilbert space of a quantum system of two particles in the infinite square well. The claim is that this is $L^2([0,1] × [0,1])$, which is said to be isomorphic to the ...
In my introductory Topology course, a completion of a metric space $(X,d)$ is a complete metric space $(\tilde{X}, \tilde{d})$ together with an isometry $i : (X, d) \rightarrow (\tilde{X}, \tilde{d})$ ...