Questions tagged [dense-subspaces]

For questions related to dense subspaces. In general topological spaces, a dense set is one whose intersection with any nonempty open set is nonempty.

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A separable normed space that is continuously embedded in a non-separable normed space implies that this embedding isn't dense.

As a preliminary I introduce the definition of denseness I am using: Definition (dense subsets of metric spaces). Suppose $(M,d)$ is a metric space. A subset $S \subset M$ is called dense in $M$ if ...
xyz's user avatar
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Necessity of Hausdorff-ness in "continuous function determined by its values on a dense subset"

It's well-known that if a continuous function taking values in a Hausdorff space is uniquely determined by its specification on a dense subset of the domain. Now, I contemplate on the necessity of ...
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The complement of any set of first category on the line is dense iff the intersection of any sequence of dense open sets is dense

I want to prove that The complement of any set of first category on the line is dense iff the intersection of any sequence of dense open sets is dense. but I don't need to prove this in general. I ...
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If two function agree on a dense set, what condition is necessary for them to be equal?

Let $f$ and $g$ be functions from $\mathbb{R}$ to $\mathbb{R}$, and $g$ a continuous function. If $f(x)=g(x)$ for every $x$ in a dense set $A$ in $\mathbb{R}$, does $f$ having the Darboux property ...
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Given $\varepsilon > 0$ and an irrational $\alpha$, there exists a rational number $m / n$ that is ($\varepsilon / n$)-close to $\alpha$?

Since $\mathbb{Q}$ is dense in $\mathbb{R}$, given a real $\varepsilon > 0$ and an $\alpha$, there exist $m, n \in \mathbb{Z}$ such that $$\left|\frac{m}{n} - \alpha \right| < \varepsilon \,.$$ ...
Henrique Fonseca's user avatar
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Topology with given infinite dense sets

Suppose $\mathcal{A}$ is a non-empty upward-closed family (i.e. $A\in \mathcal{A}$ and $A\subseteq B$ implies $B\in\mathcal{A}$) of infinite subsets on an infinite set $X$. In ZFC, does there exist ...
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Topology with given dense sets

Suppose $\mathcal{A}$ is a non-empty upward-closed family (i.e. $A\in \mathcal{A}$ and $A\subseteq B$ implies $B\in\mathcal{A}$) of non-empty subsets on a non-empty set $X$. In ZFC, does there exist ...
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intersection of sets not dense implies interior of closure of intersection is not dense

$G_n$ is a countable family of dense open sets X is a metric space I would like to show that if $\bigcap G_n$ is not dense in X then $int(\overline{\bigcap G_n})$ is also not dense in X. I have tried ...
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Prove that simple functions are not dense in weak $L^{p,\infty}$

Let $p > 0$, and denote by $L^{p,\infty}(\mathbb{R})$ the space of all measurable functions $f : \mathbb{R} \rightarrow \mathbb{R}$ for which $$ \|f\|_{p,\infty}:=\sup_{\alpha > 0} \alpha^p |\{x ...
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Lebesgue measure of complement of Q-invariant set

So I was reading up on Linear order of the quotient generated from Vitali relation implies non-measurability of subset of reals. There it was claimed either the relation or it's complement have to ...
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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 $M$?
user12345's user avatar
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Extension of closed operator densely defined

Let $X$ be a Banach space. Suppose $A:D(A)\subset X \longrightarrow X$ is a closed linear operator and $D(A)$ is dense in $X$. Prove that $A$ cannot be extended as a closed linear operator to any ...
M. Rubick's user avatar
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Showing a Certain Set of Maps is dense

Define $S=\{f\in\mathcal{C}^1([0,1]^d,\mathbb{R}^{d\times d}):\det(f(x))\text{ is tranverse to }0\}$ Another way of defining $S$ is $\{f\in\mathcal{C}^1([0,1]^d,\mathbb{R}^{d\times d}):\text{for all }...
Andrew Murdza's user avatar
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Showing a certain dense subset of $\mathcal{C}^1([0,1]^2,\mathbb{R}^4)$ is residual.

For every $f\in X\doteq \mathcal{C}^1([0,1]^2,\mathbb{R}^4)$, define $S_f=\{x\in[0,1]^2:f_1(x)f_2(x)-f_3(x)f_4(x)=0\}$. I am wondering if the set $G=\{f\in X:S_f\text{ is contained in the union of ...
Andrew Murdza's user avatar
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Nearly covering the complement of a nowhere dense set

Let $A\subseteq [0,1]$ be a nowhere dense subset of $[0,1]$ with Lebesgue measure $0$, and let $A'$ be its complement. I want to prove that, for any fixed small $\varepsilon>0$, we can find a ...
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Finitely generated dense subgroup of an unitary group

I am interested in finitely generating dense subgroups of the unitary group $U(d)$ of order $d$ (with respect to the operator norm topology), and am considering the following question. Let $I_d$ be ...
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Density of matrix functions with non-zero determinant

It is well known that the set of $A\in\mathbb{R}^{n\times n}$ such that $\det(A)\ne 0$ is dense in $\mathbb{R}^{n\times n}$. This is one way to show this: Let $B\in\mathbb{R}^{n\times n}$ with ...
Andrew Murdza's user avatar
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A dense set of real numbers with predetermined Lebesgue measure

Let $(\mathbb{R},\mathcal{M},m)$ be the set of real numbers with Lebesgue measure $m$. Fix $\epsilon\in[0,1]$. Does there exist an everywhere dense set $X\subset \mathbb{R}$ such that for any interval ...
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Is it true that the intersection of two dense subspace of a linear normed space is also dense provided that one of them is of finite codimension?

Let $X$ be a normed vector space and $X_1,X_2$ be two dense subspace of $X,$ in Is is true that the intersection of two dense subspaces of a linear normed space is also dense? it was showed by an ...
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Superfluous proof in textbook? (For intersection of sequence of dense, open subsets of complete metric space is dense)

I was wondering if my textbook's proof was superfluous. It makes use of closed balls which don't seem necessary. "If (U_n) is a sequence of dense open subsets of S, then the intersection X = (...
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Approximating Functions Limit Integral

Let $f, g \in L^1[0,1)$ such that $fg\in L^1[0,1)$. Can we find sequences $\{f_k\}\subseteq L^2[0,1)$ and $\{g_k\} \subseteq L^2[0,1)$ such that $$ \lim_{k\to\infty} \int_0^1 f_k(x)g_k(x)\,dx = \int_0^...
Doofenshmert's user avatar
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Solution set of any polynomial in $\mathbb{R}[x,y]$.

Lee in Introduction to Topological Manifolds claims that the solution set $X(f)$ of any polynomial $f\neq 0$ in two variables over $\mathbb{R}$ is nowhere dense i.e. $A:=\mathbb{R}^2-\overline{X(f)}$ ...
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Given an open set $U \subset \mathbb{R}$, is it possible to express $U$ as a disjoint union of $(a,b)$'s where $a,b \in Q$ for some dense set $Q$?

I am aware that any open subset $U \in \mathbb{R}$ may be expressed as a countable disjoint union of open intervals. Now, if we are given a dense subset $Q \subset \mathbb{R}$, is it possible to ...
Keith's user avatar
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Conditions such that complement of level set in smooth manifold is dense

Let $M$ be a smooth (hausdorff, second countable) manifold of dimension $n\geq 2$ and consider a $C^\infty$ function $f:M\to\mathbb R$. Q: Under what conditions is $S = M\setminus f^{-1}(0)$ is dense ...
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Let $f\in C^1(\mathbb{R})\cap L^2(\mathbb R)$ s.t. $f'\in L^2(\mathbb{R})$. Then approximate $f,f'$ by $f_n,f_n'$, where $f_n\in C_c(R).$

Prob. Let $f\in C^1(\mathbb{R})\cap L^2(\mathbb R)$ such that $f'\in L^2(\mathbb{R})$. Then approximate $f,f'$ by $f_n,f_n'$, where $f_n\in C_c^1(\mathbb{R}).$ Possible Hope: We know that $C_c^1(\...
CCCC's user avatar
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Is the set of polynomials of odd degree dense in $\mathcal{C}^{0}([a,b])$?

Let $$E=\{\sum\limits_{k=0}^{n}a_{k}\,x^{2k+1}\, | \, n\in\mathbb{N}\cup\{0\}\}$$ be the set of polynomials of odd degree in each term defined on $[1,2]$. (a) Show that $E$ is not closed in $\mathcal{...
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How does the reflexivity of $V$ imply the denseness of $R(i^*)$?

For a linear map $T$, we denote by $R(T)$ its range and by $N(T)$ its kernel. I'm reading about Gelfand triple at page 136 in Brezis' Functional Analysis. Let $(H, \langle \cdot , \cdot \rangle_H)$ ...
Akira's user avatar
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When can monothetic groups be turned into rings?

If G is a Hausdorff topological group, saying that G is monothetic is equivalent to saying there exists a homomorphism $f: \mathbb{Z} \to G$ with dense image. A multiplication can be naturally defined ...
Pedro Lourenço's user avatar
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Nowhere Dense Sets in Discrete Metric Space

Not Homework- Just Personal Study Let $(M,d)$ be a discrete metric space. I want to show that the only subset of $M$ that is nowhere dense is the empty set. Let $E$ be a non-empty subset of $M.$ The ...
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Nowhere Dense Set and Corollary to Baire's Theorem

Not Homework just Personal Study Reference text: Kolmogorov and Fomin - Introductory Real Analysis I am working on understanding why a complete metric space with no isolated points is uncountable. The ...
Mud's user avatar
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There are dense orbits in the set of all allowed sequences

Lately, I have done an exercise in the book "Introduction to Dynamical Systems" by Brin and Stuck. Exercise 1.4.5: Assume that all entries of some power of $A$ are positive. Show that in ...
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$x, Ax, \cdots, A^mx, \cdots$ is not dense in $\Bbb R^n$, where $A$ is a $n\times n$ real matrix, $x\in\Bbb R^n$.

$x, Ax, \cdots, A^mx, \cdots$ is not dense in $\Bbb R^n$, where $A$ is a $n\times n$ real matrix, $x\in\Bbb R^n$ are fixed. This is a problem involving mathematical analysis and higher algebra. I ...
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Prove that set of matrices is dense in $U(2)$

Consider the group of matrices $B$ generated by taking products of the matrices \begin{equation} \rho_1 = \begin{pmatrix}\exp(-4\pi i/5) & 0\\ 0 & \exp(3\pi i/5)\end{pmatrix}\\ \rho_2 = \begin{...
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Connected subgroups of unit circle

It's well known that any subgroup H of $\mathbb{S}^1$ is either dense or finite. Therefore, if H is compact, it implies that H either is finite or equal to the entire unit circle. My question is ...
Pedro Lourenço's user avatar
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Continuous map sending dense subset via a homeomorphism

This problem has been solved in and old thread. A homeomorphism on a dense set in Hausdorff space But I couldn't complete the proof so I'm asking it again here. Let $X$ be a Hausdorff space, $D \...
Its me's user avatar
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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 $\mathbb R^n$...
Daron's user avatar
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Is the cardinality of $L^1[0,1]$ greater than $\frak c$?

I am looking for a normed space whose completion has strictly larger cardinality. I have settled on the space $I^1[0,1]$ of simple functions on $[0,1]$ with completion $L^1[0,1]$ the space of ...
Daron's user avatar
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Real line and the Sorgenfrey line have the same dense subsets

I am trying to prove that the real line and the Sorgenfrey line have the same dense subsets. That is, $A\subset \Bbb R$ is dense under the lower limit topology if and only if it is dense under the ...
Raven's user avatar
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A particular covering made up by disjointed families of Closed Balls

Proposition. Let $(X,d)$ a metric space and let $A\subseteq X$ a subset of $X$ such that $A$ is separable (then $A$ has a dense -with respect the subspace topology of $A$ - and countable subset $D$). ...
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Determine if the following subspace is dense

Let $H$ be a (separable) Hilbert space, and let $\{e_n\}_{n\in\mathbb{N}}$ be an orthonormal basis. Define the following function \begin{align*} \phi:\mathbb{C}&\to H\\ z&\mapsto \sum_{n\in\...
TheEmptyFunction's user avatar
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If $H$ is dense in $\mathbb{R}$ and $\alpha\in\mathbb{R}$ then there exist a sequence $\{\alpha_n\}$ converging to $\alpha$ and $\alpha\leq\alpha_n$

If $a\in\mathbb{R}$ there exists a sequence $\{\beta_n\}\subset H$ converging to $\alpha$. Then the sequence $\alpha_n=\alpha+|\beta_n-\alpha|$ has the desired property. The problem is that I'm not ...
isaac098's user avatar
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On the size of a certain topology defined by an inclusion function.

I found this problem in chapter 7 of biglist. This is supposed to be an easy problem, but I have no idea how to approach it. The problem goes as follows: Let $(X, \mathcal{T})$ be a topological space ...
Ryukendo Dey's user avatar
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1 answer
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Density of smooth functions on $(0,T)\times\Omega$ in $W(0,T)$

Let $\Omega$ be a bounded domain and define $$W(0,T) = \{u \in L^2(0,T;H^1_0(\Omega)) : u_t \in L^2(0,T;H^{-1}(\Omega))\}$$ where $u_t$ means the weak temporal derivative. I am looking for a result ...
StopUsingFacebook's user avatar
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1 answer
57 views

Inductive limit and denseness

Let $\{E_n, n\in \mathbb{N}\}$ be an increasing sequence of linear subspaces of a vector space $E$, i.e. $E_n \subset E_{n+1}$ for all $n\in \mathbb{N}$ such that $E = \bigcup_{n\in \mathbb{N}} E_n$. ...
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Brezis' exercise 5.27

I'm trying to solve an exercise in Brezis' Functional Analysis Let $(H, \langle \cdot, \cdot \rangle)$ be a real Hilbert space and $|\cdot|$ its induced norm. Let $D$ be a subset of $H$ such that $\...
Akira's user avatar
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Is $D \setminus \{x\}$ dense?

Let $V$ be a normed linear space and $D$ be a dense subset of $V.$ Let $x \in D.$ My question is $:$ Is $D \setminus \{x\}$ dense in $V\ $? Clearly, $V \setminus \{x\} \subseteq \overline {D \setminus ...
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On showing that $T^*\text{ is unique on } H\implies\mathscr{D}(T)\text{ is dense in $H$}$

I have read that the adjoint of a linear operator $T$ on a Hilbert space $H$ is uniquely defined if and only if the domain of $T$, $\mathscr{D}(T)$ is dense in $H$. The "if" direction is a ...
Cartesian Bear's user avatar
3 votes
1 answer
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Continuity using nets on a dense subspace and regularity

If $Y$ is regular, $f:T\to Y$ is a function such that $X\subseteq T$ is dense in $T$, and for any net $x_\alpha\in X$ converging to $x\in T$, $f(x_\alpha)$ converges to $f(x)$, then $f$ must be ...
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Show that $f(E)$ is dense in $f(X)$ if $f$ continuous and $E$ dense in $X$ (Exercice $4$ from Rudin PMA chapter $4$)

Let $f$ be a continuous function defined on $X$ and let $E$ be a set dense in $X$. We aim to show $f(E)$ is dense in $f(X)$. We know that: $f(\overline{E})\subseteq \overline{f(E)}$ for every ...
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2 votes
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Bijective extension of Lipschitz map defined over a dense set

Let $Y$ be a complete metric space (I'm interested in particular in the case $Y=c_0$, if relevant) and $X\subset Y$ dense. Given a 1-Lipschitz bijection $f$ from $X$ onto $X$, does it extend to a 1-...
Stefano Ciaci's user avatar