Mathematics

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.

Given a metric space $(X,\rho)$, we say that a subset $A$ of $X$ is dense in $X$ if, for each $\epsilon >0$ and $x\in X$ $$\tag 1 B(x,\epsilon)\cap A\neq \varnothing$$

This can be put more succinctly as ${\rm cl}\; A=X$. That is, the closure of $A$ is $X$.

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Is there a path-connected, "anti-convex" subset of $\mathbb R^2$ containing $(\mathbb R\smallsetminus \mathbb Q)^2$?

This question was firstly asked in Mathematics Stack Exchange. Getting no answer, I copied it to Math Overflow, as Moishe Kohan commented. For a vector space $V$ over $\mathbb R$ , I say a subset $S$ of $V$ is "anti-convex" if $\forall a,b\in S…
yummy
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Any counter-example to Weierstrass-Stone Thm on non-compact spaces

I would like to show, out of curiosity really, that Weierstrass-Stone Thm fails when the underlying space is not compact. Specifically, I think that $C_b(\Bbb R), $ the space of bounded continuous functions on $\Bbb R $ should suffice. To this…
Maurice
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Open dense subset of a compact lie group has full measure

For a given compact (lie) group there exists a unique haar propability measure $\mu_G$ . Is it true that any open dense subset $U\subseteq G$ has full measure, i.e. $\mu_G(U) = 1$ ? I know the statement is wrong for locally compact groups, see e.g.…
Tina
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Density in Besov spaces $B_{p\infty}^{s}\left(\mathbb{R}\right)$ and $B_{\infty \infty}^{s}\left(\mathbb{R}\right)$

Let $0<s<1$ . I'm interested to find a class of smooth functions such that they are dense in $B_{p\infty}^{s}\left(\mathbb{R}\right)$ with $1 \leq p <+\infty$ with respect the norm $$ \left\Vert f\right\Vert…
Yep
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Pointwise convergence on a dense subset of the domain

Let's say that $X$ and $Y$ are vector spaces and to be more accurate $Y$ is a dense subspace of $X$. Furthermore we have that $(f_n)_{n\in\mathbb{N}}$ is a sequence in $X^*$, $f\in X^*$ and that $f_n$ converges pointwise to $f$ on $Y$. Can we deduce…
Perpendicular
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Uniform boundedness principle on dense subspace

In my thesis I encountered the following problem: Let $X,Y$ Banach spaces and $Z\subset X$ a norm-dense subspace. Suppose we have operators $\left\{T_n:n\in\mathbb{N}\right\}$ such that for all $x\in Z$ it holds that $\left\{\left\lVert…
WillemMSchouten
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Weakly*-Dense Subspaces of $\ell^\infty$

Identify $\ell^\infty=(\ell^1)^*.$ I'm interesting in whether there exists a closed, weakly $^*$ -dense subspace $F$ of $\ell^\infty$ s.t. $c_0\not\subseteq F.$ My hope is that the answer is no, but I concocted the following space. It is clearly…
Miles Gould
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Dense subring of composite of valued fields

Let $(F,v)$ be a valued field, $(K_1,v\vert_{K_1})$ and $(K_2,v\vert_{K_2})$ be two complete valued subfields of $(F,v)$ . Let $R_1$ (resp. $R_2$ ) be a dense (w.r.t. to the topology induced by $v$ ) subring of $K_1$ (resp. $K_2$ ). My question is:…
Yijun Yuan
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Understanding Proof: Simple Functions In $\mathscr{L}^{\infty}(X,\mathscr{A},\mu)$ Form A Dense Subspace of $\mathscr{L}^{\infty}(X,\mathscr{A},\mu)$

I have difficulties understanding the proof of the following proposition: Proposition $\quad$ Let $(X,\mathscr{A},\mu)$ be a measure space, and let $p$ satisfy $1\leq p$ . Then the simple functions in $\mathscr{L}^{\infty}(X,\mathscr{A},\mu)$ form a…
Beerus
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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\left(\dfrac{-4\pi i}{5}\right) & 0\\ 0 & \exp\left(\dfrac{3\pi i}{5}\right)\end{pmatrix}\\ \rho_2 =…
PhPanda
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Making dense subset of continuous functions using a single continuous function

I have a question about set of continuous function on the compact interval. Denote the set of all continuous $n$ -dimensional real functions on $[0,L]$ as $\mathcal{C}_{[0,L]}$ ( $\mathcal{C}_{[0,L]} = \left\{ u: [0,L] \rightarrow…
박희인
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Is the set $S=\{ p(x) e^{-\alpha \|x\|^2} : p(x) \in \mathcal{P}, \alpha >0 \}$ dense in $L^2$ where $\mathcal{P}$ is a set of polynomials.

Let $\mathcal{P}$ be the set of all realvalued polynomials on $\mathbb{R}^2$ and define $$S=\{ p(x) e^{-\alpha \|x\|^2}, x\in \mathbb{R}^2 : p \in \mathcal{P}, \alpha >0 \}.$$ I have two questions, one minor and one major: (Minor) Is $S$ is a dense…
Boby
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Are Trigonometric Functions Dense in $C^k(S^1)?$

Consider the functions $\{e^{2\pi i nx}\}_{n \in \mathbb{Z}}$ defined on the interval $[0,1].$ These are all smooth periodic functions (so functions on $S^1)$ and by the Stone-Weierstrass theorem they are dense in $C^0(S^1)$ when it is given the max…
mck
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Preduals of a Banach Space vs Minimal Closed, Weakly$^*$-Dense Subspaces of its Dual

Let $X$ be a Banach space. Define the ordering $\leq$ on the subspaces of $X^*$ by $Y\leq Z$ iff $Y$ is a closed subspace of $Z.$ Then we define the following collections: $$\mathrm{Du}(X)=\{Y\leq X^*|Y\text{ weakly}^*\text{-dense in }X^*\}$$ the…
Miles Gould
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Direct proof for existence of dense set in topological space.

I am working on Mendelson's Introduction to topology. This question arose after completing question $8$ from the exercise on page $87.$ A subset $A$ of a topological space $(X,\mathfrak{I})$ is said to be dense in $X$ if $\overline{A}=X.$ Prove that…
Adam Rubinson
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