Questions tagged [complete-spaces]

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

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Show that $\{x\in\mathbb{C}^{\mathbb{N}}: \lim_{n\to\infty} \frac{x_n}{a_n}=0\}$ is a Banach space

Consider $a:=\{a_n\}_{n=0}^{\infty}\in\mathbb{R}^{\mathbb{N}}_{+}$ a decreasing sequence with $a_n\to 0$. Consider the following space of sequences $$ X_a:=\left\{x:=\{x_n\}_{n=0}^{\infty}\in\mathbb{C}...
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Show that if $(X,p)$ is complete, then so is $(X,d)$ as follows.

Let $d$ and $p$ be two metrics on $X$ such that $$\frac{1}{2022}p(x,y) \le d(x,y) \le \frac{p(x,y)}{p(x,y)+1},$$ for all $x,y \in X$. Show that if $(X,p)$ is complete, then so is $(X,d)$. Attempt: Let ...
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Is the following metric space complete?

Let $X=\{x=(x_i)_{i \in \mathbb{N}} \in \mathbb{R}^{\mathbb{N}} \ \vert \ \exists N \in \mathbb{N} : x_i \geq 0 \ \ \forall i \geq N\}$ and let $\bar{\rho}$ be the uniform metric on $\mathbb{R}^{\...
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How to prove that every Cauchy net in a complete metric space is convergent? [duplicate]

How to prove the basic fact that every Cauchy net in a complete metric space is convergent? My definition: A metric space $X$ is complete iff every Cauchy sequence in $X$ is convergent. I don't ...
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Can we guarantee the completness of a closed subset of a complete metric space?

As a second question: can we guarantee the completness of (the subset as subspace) a open subset $L$ of a complete metric space $M$ ? Complete subspace $L$, of metric space $M$, is a closed set in $...
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-1 votes
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Is the field of complex numbers complete? [closed]

I know that the complex numbers can't be ordered, does this imply that the field $\mathbb{F}$ of $\mathbb{C}$ is not complete? do we need any other proof other than what I stated? Can we use the same ...
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If $X$ is complete separable, then the space $\mathcal{P}(X)$ of all Borel probability measures on $X$ is separable in Prokhorov metric

I'm trying to prove below result. Could you verify if my attempt is fine? Let $(X, d)$ be a metric space and $\mathcal{P} :=\mathcal{P}(X)$ the space of all Borel probability measures on $X$. Let $d_P$...
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How to prove the space $L^2\big([a,b],w(t)\big)$ is complete and separable?

Suppose $w(t)$ is a positive and measurable function on $[a,b]$. If $x(t)$ is a measurable real function on $[a,b]$ satisfying $$ \left\| x \right\|^2 =\int\limits_{\left[ a,b \right]}{w\left( t \...
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If either $\mathcal M(X)$ or $\mathcal P(X)$ is complete, then so is $X$

I'm trying to prove this result from Wikipedia. Could you have a check on my attempt? Let $(X, d)$ be a metric space. Let $\mathcal{M} :=\mathcal{M}(X)$ and $\mathcal{P} :=\mathcal{P}(X)$ be the set ...
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Zernike polynomials, Bessel functions

Are Bessel functions an equivalent alternative to represent a complete set (basis of vector space) on the unit disk? As Zernike polynomials are an orthonormal basis, I wonder if a similar property ...
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Are compositions of functions from compact function spaces compact?

Let $X\subset \mathbb{R}^d$ and $A\subset \mathbb{R}$ be a compact sets, where $C_B(X)$ and $C_B(A)$ denote the space of continuous and bounded real-valued functions $X\rightarrow\mathbb{R}$ and $A\...
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Completeness of a metric with discrete topology

Let $\mathbb{N}$ be a set of natural numbers. Define a metric $d$ as follows, $$d(m,n)=\left |\frac{m-n}{mn} \right | \quad for \: all \: m,n \in \mathbb{N}.$$ With this metric, every singleton subset ...
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Countable product of complete spaces is complete

Consider Fréchet spaces $\{E_n\}_{n\in\mathbb{N}}$, and let $E = \prod_{n=1}^{\infty} E_n$. Show that $E$ is a Fréchet space. I'm stuck with proving completeness. I have already verified that $E$ is ...
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Characterization of $\mathbb{R}$ as being a complete Riemannian manifold

This is the exersise 13-16 of Lee's Introduction to Smooth Manifolds. Let $R$ be a riemannian manifold equiped with the metric $g = f(t) dt^2$. I want to show that $(\mathbb{R},g)$ is complete (in the ...
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Completing a metric space where only (eventually) positive, negative or zero sequences converge.

Let $X = \mathbb{R}$ and for $x, y \in \mathbb{R}$ define $d'(x, y)$ by $d'(x, y) = \begin{cases} |x - y|+1 & if \space exactly \space one \space among \space x\space and\space y\space is\...
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What's the easisest way to show that this $L^2$-like space is complete?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probaiblity space, $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$ and $E$ be a normed $\mathbb R$-vector space. If $t\ge0$ and $(X_s)_{...
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How can I show that the Sobolev space $W_p^1 (a,b)$ is closed?

We define the Sobolev space $W_p^1 (a,b)$ as: $$W_p^1 (a,b) = \left\{ u \in L_p (a,b) \ : \ u \in AC\left[a, b\right], \ u' \in L_p \left[a, b\right] \right\}$$ where $AC\left[a,b\right]$ means "...
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Proof check regarding complete space with seminorm

I just came across a simple Functional Analysis exercise which I just dont know if I correctly solved. The problem states: Consider the space $C(\mathbb{R}):=\{f: \mathbb{R} \longrightarrow \mathbb{C};...
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Proving that the sets $P_n = \{x\in X\mid |f(x)| \leq n, \forall f \in A\}$ have empty interiors

Let $(X, d)$ be a complete metric space and $A$ a set of continuous functions $f:X\to \mathbb{R}$ s.t. the sets $S_x = \{f(x)\mid \forall f \in A\}$ are bounded for each $x \in X$. Define $P_n = \{x\...
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Does there exists a homeomorphisms which are not isometries,but preserves completeness??

Does there exists a homeomorphisms which are not isometries,but preserves completeness?? The above question clicked my mind while studying the relations $1)$Isometry$\implies$ Homeomorphism$\implies$ ...
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Proof explanation (Complete subspace)

While studying functional analysis, more specifically that the subspace $Y=\{x \in \mathcal{C}[a, b] \mid x(a)=x(b)\} \subset \mathcal{C}[a, b]$ is complete, I came across a very simple question I ...
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Help understanding why this sequence converges

I am starting to studying Functional Analysis through Kreyszig's book. Right now I am studying this proposition: Theorem (Complete subspace). A subspace $M$ of a complete metric space $X$ is itself ...
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Constructing completion of metric space on closed intervals

Given $$X=\{[a,b] | -\infty < a < b < \infty \}$$ and $|I|=b-a$ for $I=[a,b]$, I have already proved that $d: X\times X \to \mathbb{R}$ where $$d(I,J)=|I|+|J|-2|I\cap J|$$ is a metric. I am ...
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Understanding proof: compact iff complete and precompact

Let $E$ be a Hausdorff topological vector space. Then $K\subseteq E$ is compact if and only if $K$ is complete and precompact (i.e., the closure of $K$ in the completion $\hat{E}$ of $E$ is compact). ...
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The unit disk with pseudo hyperbolic distance is complete space

The pseudo-hyperbolic distance on the unit disk $D$ is defined as: $$\rho(z,w)=\left|\dfrac{z-w}{1-\bar wz}\right|.$$ I need to prove that $(D,\rho(z,w))$ is complete space. I know that a metric space ...
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4 votes
1 answer
116 views

Show that $(Y,||\cdot||_Y)$ is a Banach space

Let $(X,||\cdot||_X)$ be a Banach space and $(e_n)_{n\in\mathbb{N}}$ a Schauder basis of X. How can I prove that $Y:=\{\alpha:\mathbb{N}\to\mathbb{R} \space| \lim_{N\to\infty}\sum_{n=0}^N\alpha_n e_n \...
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1 vote
2 answers
31 views

Isomorphism between $\mathbb{C}^{n}$ and the Complexification of $\mathbb{R}^{n}$

What is an easy way (without tensors) to show, that the complexification of the Euclidean space $\mathbb {R} ^{n}$ gives the unitary space $\mathbb {C} ^{n}$. Or more detailed: I'd like to show, that $...
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Baire-1 functions properties on a complete metric space $(X, d)$

Let be $(X, d)$ a complete metric space, $f_n : X \rightarrow \mathbb{R}$ a sequence of continuous functions and a function $f: X \rightarrow \mathbb{R}$ such that: $$\lim_{n \rightarrow +\infty} f_n(...
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Can we have complete lattice "inside of" something that is not a lattice?

Can we have complete lattice "inside of" something that is not a lattice? Say we have a partially ordered set $P$. Can we have a complete lattice $Q$, which has ordering inherited from $P$, ...
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4 votes
1 answer
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Cauchy sequences , continuity and completeness

We have been taught that if $X$ and $\mathbb{R}$ are two complete metric spaces then $f:X \to \mathbb{R}$ is a continuous function iff it carries cauchy sequences to cauchy sequences. However let us ...
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$e^{i2\pi a}=1$ or $e^{i2\pi a}=\cos(2\pi a)+i\sin(2\pi a)$? [duplicate]

Since $e^{i2\pi a}=(e^{i2\pi})^a=(1)^a=1$ but $e^{i2\pi a}=\cos(2\pi a)+i\sin(2\pi a)\neq1$, why is that?
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Can an inner product space contain a complete orthonormal sequence, and still be incomplete?

Can an inner product space contain a complete orthonormal sequence, and still be incomplete? This question came to my mind while reading Optimization by Vector Space Methods by David G. Luenberger. ...
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Alternative definition of equivalent Cauchy sequences in a metric space

Suppose $M$ is a metric space, not necesserily complete. Given the following definition of Cauchy sequence: For every $\epsilon > 0$ there exists an open $\epsilon$-ball that contains all but a ...
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3 votes
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Examples of Cauchy complete ordered fields that are not $\mathbb{R}$?

According to this post, it is not true that Cauchy completeness (every Cauchy sequence has a limit) and Dedekind completeness (every nonempty set that is bounded above has a supremum) are equivalent ...
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Metric on $[0,1)$ that is not complete nor totally bounded

I am looking for a metric on $[0,1)$ that is not complete nor totally bounded. $[0,1)$ with the Euclidean distance is not complete but totally bounded, other norms are equivalent. With the discrete ...
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Completeness of the Metric Space $2^A/\sim$ [duplicate]

Let $A$ be some elementary subset of $\mathbb{R}^d$, and define an equivalence relation on $2^A$ by declaring two sets $E,F\subset A$ equivalent if $m^*(A \triangle B) = 0$, where $m^*$ is Lebesgue ...
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2 votes
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Is it enough to find $g$ not in $C([a,b])$ such that $f_{n}\rightarrow g$ in $||\cdot||_{p}$to conclude $(C[a,b],\|\cdot\|_p)$is not complete?

I have some questions about the Danny Pak-Keung Chan 's answer given in this post $C[a,b]$ is not Banach under Lp norm to prove $C[a,b]$ is not complete with the p-norm. I am copying the answer and ...
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Is the function space $X=\{u \in H^1(\Omega) : \text{$u$ is continuous at $0$}\}$ complete?

This question is somewhat similar to Form functions that are continuous at one point in L^\infty a Banach space. where the continuity at zero was added to $L^\infty(\Omega)$. This space was indeed ...
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1 vote
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Is the orthogonal complement of a subspace in a Hilbert space always complete?

I think this is true but I am not sure because I can't find it written in any book I have read so far and this seems to be an important result. Here is my argument: Let $M$ be a subspace of an inner ...
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Why the space of section of a vector bundle is complete?

This page is about the space of sections: Let $E \stackrel{\mathrm{fb}}{\rightarrow} \Sigma$ be a smooth vector bundle. On its real vector space $\Gamma_{\Sigma}(E)$ of smooth sections consider the ...
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Show that $H^{1}(\mathbb{R}^n)$ is properly included in the completion of $C_{c}^{\infty}(\mathbb{R}^{n})$

Let $X$ be the vector space completion of $C_{c}^{\infty}(\mathbb{R}^n)$ under the norm $$ \|u\|_{X}=\left(\int_{\mathbb{R}^{n}}|Du|^{2}dx\right)^{\frac{1}{2}}. $$ I have shown that $H^{1}(\mathbb{R}^...
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5 votes
3 answers
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Proof that the space of vector valued function is complete

Let $\Omega \subseteq \mathbb{R}^{2}$ be open let $C^{\infty}(\Omega, \mathbb{R}^{2})$ denote the space of all smooth functions from $\Omega$ to $\mathbb{R}^{2}$. We define a norm in $R^2$ given ...
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1 vote
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Examples of incomplete normed spaces of continuous linear maps between two normed spaces [duplicate]

Let $X,Y$ be normed spaces, $B(X,Y)$ the normed space of continuous linear maps between $X$ and $Y$. Are there normed spaces $X,Y$ such that $B(X,Y)$ is not complete? It is well-known that $B(X,Y)$ is ...
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1 vote
1 answer
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How to show the completeness of the following subset of $\ell^2$?

Let's define the subset of $\ell^2(\mathbb C)$ $$\mathcal D(A) = \left\{ {z \in {\ell ^2}\left( C \right),\sum\limits_{k = 1}^\infty {k^2{{\left| {{z_k}} \right|}^2} < \infty } } \right\},$$ I ...
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1 vote
1 answer
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Uniformly continuous function on totally bounded subset of complete metric space

I have the following problem at hand: Let $(X, d)$ be a complete metric space and $f: X \rightarrow Y$ be continuous. Prove that $f$ is uniformly continuous on any totally bounded subset of $X$. My ...
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0 votes
1 answer
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Convergence of fixed points of a sequence uniformly convergent.

Let $(M,d)$ a metric space, a function $f:M\to M$, is a contraction if there exists $k\in [0,1)$, such that $d(f(x),f(y))\leq k\,d(x,y)$. If $(M,d)$ is complete, one can show that $f$ has a unique ...
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2 votes
1 answer
122 views

Proving that the metric space $(\mathcal{H}(\mathbb{D}), d)$ is complete using Complex Analysis

Call the vector space of holomorphic functions on the unit disk $\mathcal{H}(\mathbb{D})$. Define for $f, g \in \mathcal{H}(\mathbb{D})$: $$d(f,g) = \sum^{\infty}_{k=1}2^{-k}\min\left(1, \sup_{|z|\...
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3 votes
1 answer
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Prove that $\Phi$ is a lower semicontinuous functional.

QUESTION: Let $(X, \|.\|)_X$ be a Banach space and $Y\subset X$ a subspace, which is itself a Banach space endowed with a norm $\|.\|_Y$ such that $\|y\|_X\leq \|x\|_Y$ for every $y\in Y$. Assume that ...
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3 votes
1 answer
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Prove that $\Gamma=\{\gamma:K\longrightarrow (Y, \| . \|) : \gamma\; \text{is continuous and} \; \gamma|_{K_0}=\gamma_0\}$ is a complete metric space.

QUESTION: Let $(X, \|.\|)_X$ be a Banach space and $Y\subset X$ a subspace, which is itself a Banach space endowed with a norm $\|.\|_Y$ such that $\|y\|_X\leq \|x\|_Y$ for every $y\in Y$. Let $K$ be ...
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  • 399
1 vote
0 answers
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Necessary and suficient condition for a uniform space to be complete

I am trying to learn a bit about linear topological rings and modules. In doing that, I have stumbled upon some questions about uniform spaces and I am struggling to find references. My main is ...
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