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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Let $(X,d)$ and $(Y,\rho)$ be two metric spaces with a bijection between them. If $(X,d)$ is complete then can we say $(Y,\rho)$ also complete?
The question arise when I study complete space . We know if there a bijection between two set then their property will same ,is it true for completeness property of a metric space. We know $\mathbb{R^...
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Finite index dense normal subgroups of completely metrizable groups
Is there some completely metrizable group $M$, which contains a normal subgroup $N\trianglelefteq M$ of finite index (at least $2$) that is dense in $M$?
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completeness of real and convergence of binary series
In page 'Completeness of the real numbers' of Wikipedia is said that: ''
completeness (of real) is equivalent to the statement that any infinite string of decimal digits is actually a decimal ...
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Open normal subgroups with finite image under extensions
Let $A$ be a (discrete) countable group such that for ever completely metrizable group $M$ and any (not necessarily continuous) homomorphism $f\colon M \to A$ there exists some open normal subgroup $N\...
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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 ...
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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 ...
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A question for uniform convergent of sequence of functions.
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 &...
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vector space complete in coarse norm but incomplete in finer norm
I would like to see an example of the following situation.
It is well known that $C([0, 1])$ is complete under the $\|\cdot\|_\infty$ norm, but incomplete under the $\|\cdot \|_1$ norm.
Since clearly ...
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Banach's Fixed Point Theorem application
I'm trying to solve the following question:
Consider $M$ a complete metric space, $k > 1$ and $f: M \to M$ a surjective function, satisfying $d(f(x),f(y)) \geq k d(x,y)$, for every x,y $\in M$. ...
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Prove that the space of coefficients of convergent sequences in a Banach space is a Banach space.
came across this problem in a book I'm reading:
Let $\mathbb{X}$ be a Banach space. For a sequence of nonzero vectors $(x_k)_{k=1}^{\infty}$ in $\mathbb{X}$, define $\mathbb{E}:=\{a=(a_k)_{k=1}^{\...
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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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Is space complete and separable?
$\displaystyle M =\{x\in\mathbb{R}^{(0,1]} : x \text{ is continuous on } (0,1] \text{ and } x(t) = o(\ln{}t)\ (t \rightarrow +0)\} $. We will consider metric $\displaystyle\rho(x, y) = \int\limits_{0}^...
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A distance function on a Riemannian manifold which can be viewed as a distance function to the point at infinity.
$\mathbf {The \ Problem \ is}:$ Let $f:\mathbb{R^2}\to \mathbb{R}$ be defined by $(x,y)\mapsto \frac{1}{\sqrt{2}}(x+y).$ Find a sequence $(p_i)_{i\in \mathbb {N}}$ of points in $\mathbb{R}^2$ so that $...
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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 ...
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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 $f_n(x)=x^n$. ...
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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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On Elkik's Theorem of Henselian pair.
As far as I know, the celebrated Elkik's result often quoted boil down as follows$\colon$ Let $A$ be a heneselian local ring with its maximal ideal ${\frak m}$ and choose an arbitrary ideal $I \subset ...
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Completion of operator space
Definition: An (abstract) operator space is a linear space $X$ together with a sequence $\{\|\cdot\|_n\}_{n=1}^\infty$ of norms such that
$\|x\oplus y\|_{m+n}= \max\{\|x\|_m, \|y\|_n\}$ for $x\in M_m(...
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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 ...
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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 ...
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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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Are Cauchy sequences for Hilbert space an expression of a compact (multiplication) operator?
Background: I'm reading about Hilbert spaces that require a complete metric space using inner product, where every Cauchy sequence of points $x_m$,$x_n$ on the metric space, $X$, has a limit, also in $...
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Is $l^\infty$ with this norm a Banach space?
We define for $l^\infty$ (bounded sequences) the following norm (the $n$-th term of the sequence $x$ is $x(n)$):
$$
||x||=\sum_{n=1}^{\infty}\frac{|x(n)|}{2^n}
$$
The question is if $(l^\infty, ||\...
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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 ...
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$X$ 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 ...
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A generalization Banach fixed point theorem
Here is the poblem: Let $(X,d)$ be a complete metric space and $\omega:\mathbb R_{\geq0}\to\mathbb R_{\geq0}$ is a right-continuous function such that $\omega(0)=0$ and for any $t>0$, one have $0\...
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Show that $(I([0,1]), d_H)$ is complete.
I want to show that the metric space $(I([0,1]), d_H)$ is complete, where $I([0,1])$ is the set of closed intervals $[a,b]$, $0 \leq a \leq b \leq 1$, and $d_H$ is the Hausdorff distance.
I've tried ...
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I don't understand one of the steps in proving that a Uniformly Continuous Function can be extended in a metric space.
I was reading this post regarding the Extension of a Uniformly Continuous Function between Metric Spaces. I understood how you're supposed to extend the function and why said extension is well defined ...
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Are pseudometric completions canonical in any sense?
Let $P$ be a pseudometric space, and let $M_P$ be the metric space obtained by quotienting out points separated by zero distance. We can always complete $M_P$ to $\overline{M_P}$ by forming the ...
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Infinite intersection of closed Hilbert subspaces whose finite intersection is infinite-dimensional
I'm working in the setting of complex, separable, infinite-dimensional Hilbert spaces and I came out with the following statement which, intuitively, seems to be true to me, though I haven't found a ...
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Does the SVD work on an incomplete field?
Does the singular value decomposition (SVD) require a complete field? SVD clearly can't work on $\mathbb Q$, since we need square roots. But can it work on $\mathbb E$, the smallest Euclidean field ...
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Optimal basis set of Gaussian functions for describing a quantum system
Copy from here https://mathematica.stackexchange.com/questions/285044/optimal-basis-set-of-gaussian-functions-for-describing-a-quantum-system
This question arose during the discussion of the previous ...
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How to check if a locally convex space is complete?
I am given the definition that a locally convex space $V$ is complete if and only if it is isomorphic to its completion $\tilde{V}$.
The completion is constructed in the following way: We have a ...
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A corollary of Baire Category Theorem
I have learnt that the theorem says:
Let $\{U_n\}_{n=1}^\infty$ be a sequence of dense open subsets of a complete metric space X.
Then $\displaystyle\cap_{n=1}^\infty U_n$ is also dense in X.
I also ...
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Why does the minimum eigenvalue change dramatically when one basis function is added to the basis set?
Copy from here https://mathematica.stackexchange.com/questions/284809/why-does-the-minimum-eigenvalue-change-dramatically-when-one-basis-function-is-a
I have a basis set which describes with high ...
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How to expand the existing basis set so that it becomes more complete?
This question has some mathematical aspects https://mathematica.stackexchange.com/questions/284679/how-to-expand-the-existing-basis-set-so-that-it-becomes-more-complete , so I have decided to ...
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Constructing a Homeomorphism with the Cantor Set
This question is about a blog post by Terrence Tao showing that the real line $\mathbb{R}$ cannot be expressed as a disjoint union of countably many closed intervals. There are many proofs for this ...
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complete manifolds in product metric
Let $M=\mathbb{R}\times\mathbb{S}^{n-1}$ be equipped with the warped
product metric $g(x,\theta)=dx^2+e^{-x^2}h_\theta$ , where $h_\theta$
is standard metric on $\mathbb{S}^{n-1}$ . Prove that $(M,g)$ ...
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Is this pre-Hilbert space complete?
We are given a Hilbert space $$ B(\mathbb{C})= \{ f:\mathbb{C}\to \mathbb{C}\mid f\text{ is holomorphic and }\int_{\mathbb{C}}|f(z)|^2e^{-|z|^2}dz < \infty \}$$
The inner product is given by
$$\...
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Does a completely metrizable space admit a compatible metric where all intersections of nested closed balls are non-empty?
It is well-known that given a metric space $(X,d)$, the metric is complete if and only if every intersection of nested (i.e. decreasing with respect to inclusion) closed balls of vanishing diameters ...
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A exercise assigned in the topology course
Let $\{x_n\}_{n=1}^{\infty}$ be an enumeration of $\Bbb Q$, $\{y_\lambda\}_{\lambda \in \Lambda}$ an enumeration of $\Bbb R\setminus\Bbb Q$.
Proof: for any families $\{r_n\}_{n=1}^\infty,\,\{ \rho_\...
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