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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 (a\ne…
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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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: 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…
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…
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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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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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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Linear span of $\{e^{-n x}\}_{n\in\mathbb N}$ is dense in $L^2((0,\infty))$. Explicit expansion?

I proved that the linear span of $\{e^{-n x}\}_{n\in\mathbb N}$ is dense in $L^2((0,\infty))$ (see below). Question: Given $f\in L^2((0,\infty))$, can I find "expansion coefficients" $f_n$ such that $$f(x) = \sum_{n\in\mathbb N} f_ne^{-n x}…
Laplacian
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Is it possible to define a topology on the real line such that 0 and non-zero integers are dense but no finite subset of non-zero integers is dense?

Is it possible to define a topology $\mathcal{T}$ on $\mathbb{R}$ such that $\{0\}$ and $\mathbb{Z}^*=\mathbb{Z}\setminus\{0\}$ are dense but no finite $F\subseteq \mathbb{Z}^*$ is dense? This is what I know. I know that $\{0\}$ nor $\mathbb{Z}^*$…
K. Makabre
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Maximal Non-Hausdorff Compactification

I have recently started to read about compactifications of topological spaces, however I would like to clear my mind on a few things. For starters, I am interested in generic topological spaces (not necessarily Tychonoff) although I imagine that not…
AlienRem
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