# Questions tagged [complete-spaces]

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

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### The closure of $k(t)$ with respect to $v_0$ is $k((t))$

I am trying to prove the result in the title. I have that $v_0$ is defined by $v_0\left(t^n \frac{f(t)}{g(t)}\right)=n$, where $n \in \mathbb{Z}$ and $f,g \in k[t]$ with $f(0),g(0) \neq 0$. I am not ...
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### 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 ...
1 vote
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### 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 ...
1 vote
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### Check if the norm space is complete

Question Let $X$ consist of all real values functions $f$ on $[0,1]$ such that $f(0)=0$, $\|f\|=\sup \left\{\frac{|f(x)-f(y)|}{|x-y|^{1/3}}:x\neq y\right\}$ is finite. Prove that $\|\cdot\|$ is a ...
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### In a complete space $X$ is every $x \in X$ the limit of a sequence $\{x_n\}$ such that $x \not\in \{x_n\}$? [closed]

Let $X$ be a complete metric space. Then for any point $x \in X$, can it be shown that there exists a sequence $\{x_n\} \in X$ such that $x \not\in \{x_n\}$ and $x_n \rightarrow x$? More generally, I ...
1 vote
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### The completion of metric spaces with bounded sequences instead of Cauchy sequences

Out of idle curiosity, I was mulling the idea of the completion of a metric space. In a nutshell, one starts with a metric space $(M, d)$, defines an equivalence relation $\sim$ on the set of Cauchy ...
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### Show that $C([0, 1])$ equipped with $\Vert f\Vert_1:=\int_0^1|f(x)|dx$ is not complete

Show that the normed space of continuous real valued functions $C[0,1]$ equipped with the norm $\Vert f\Vert_1:=\int_0^1|f(x)|dx$ is not complete. Let be $f_n:[0,1]\to\mathbb{R}$ with $f_n(x)=x^n$. ...
1 vote
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### $(M,d)$ complete metric space and $f : M \longrightarrow M$ such that $f^p$ is a contraction. Then, $lim f^n(x) = a$, for any $x \in M$.

I try to solve this problem: "Show that if $(M,d)$ is a complete metric space and $f: M \longrightarrow M$ is a map such that exists $p \in \mathbb{N}$ for which $f^p$ is a contraction, then, $f$ ...
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### Why is $C([0;1])$ with the supremum metric complete, but it is not with the integral metric

I am currently studying metric spaces in my mathematical analysis course and I came across two examples: First - show that the set of continuous functions on a closed interval (denoted as $C([a;b]$) ...
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### Question in Construction for Proof of Nested Sphere Theorem

Not Homework Just Personal Study: Statement of Theorem: A metric space $(M,d)$ is complete if and only if every nested sequence of closed balls $S_n=\{B(x_n,r_n)\}$ with $\lim_{n\to \infty}r_n=0$ ...
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### Prove that the limit of the sequences corresponding to the image of two sequences converging to the same point is the same.

Context : I consider $(E, d_E)$ and $(F, d_f)$ two metric spaces and $A\subset E$ a dense subset of $E$ (i.e $\bar{A}=E$). The function $f$ is defined only on $A$. I would like to prove that if I have ...
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### Boundedness of Linear Operators on Banach Subspace with Different Norm

I had this exercise on a functional analysis exam but I was unable to solve point iii). I solved points i), ii), and iv). I solved i) with the open map theorem, ii) using i) and iv) as an instance of ...
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### Range of a function on subintervals [duplicate]

Prove or disprove the existence of a function $f:[0,1] \rightarrow[0,1]$ with the following property: for any interval $\,(a,b)\subset[0,1]\,$ with $\,a\!<\!b,f\big((a,b)\big)\!=\![0,1]\,.$ It ...
1 vote
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### The metric $\hat \rho (f, g) := \inf_{\delta >0} \{ \mu (|f - g| > \delta) +\delta \}$ on the space of $\mu$-measurable functions is complete

Below we use Bochner measurability and Bochner integral. Let $(X, \mathcal A, \mu)$ be a complete finite measure space, $(E, | \cdot |)$ a Banach space, $S (X)$ the space of $\mu$-simple functions ...
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### The metric space $(L^0 (X), \rho)$ of $\mu$-measurable functions is complete

Below we use Bochner measurability and Bochner integral. Let $(X, \mathcal A, \mu)$ be a complete finite measure space, $(E, | \cdot |)$ a Banach space, $S (X)$ the space of $\mu$-simple functions ...
1 vote
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### Is there a metric $d$ such that $(\mathbb{F}[x], d)$ is complete?

Denote $\mathbb{F}[x]$ (where $\mathbb{F} = \mathbb{R}$ or $\mathbb{F} = \mathbb{C}$) is the space consisted of polynomials with coefficients in the field $\mathbb{F}$. The question is whether there ...
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1 vote
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### The space $L^p_{\text{loc}} (\mathbb R^d)$ is complete w.r.t. the norm $\|f\|_{\tilde L^p} := \sup_{x \in \mathbb R^d} \|1_{B(x, 1)} f\|_{L^p}$

Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let $\tilde L^p (\mathbb R^d)$ be the space of ...
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### Does there exist a normed space over $\mathbb R$ whose completion has strictly larger cardinality?

It is easy to come up with a metric space whose completion has strictly larger cardinality. Something like $\mathbb Q$ with completion $\mathbb R$ will do. Or more generally any subset of $\mathbb R^n$...
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### Does there exist a Cauchy-complete, non-Archimedean ordered field that is not isomorphic to a field extension of $\mathbb{R}$?

Does there exist a metrizable non-Archimedean ordered field $\mathbb{F}$ that is Cauchy-complete under some metric $d$, where $\mathbb{F}$ is not isomorphic to any field extension of $\mathbb{R}$? I ...
### $X$ is a Banach space for the norm $\|u\| := \sup_{t \ge 0} e^{-kt} |u(t)|$
Let $(E, |\cdot|)$ be a Banach space, $k>0$ and $$X := \{u \in C([0, \infty); E) : \sup_{t \ge 0} e^{-kt} |u(t)| < \infty\}.$$ In the proof of Theorem 7.3 from Brezis' Funtional Analysis, the ...