# Questions tagged [complete-spaces]

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

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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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 ...
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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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### 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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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 ...