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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Dense inclusions for $L^{\infty}$-Bochner spaces
Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. It is known that
$ L^{\infty}(\Omega)$ is dense in $L^{1}(\Omega)$ in the case that $\Omega$ is bounded, since $C_c^{\infty}(\Omega)$ is dense in ...
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Is this set dense in the space of continuously differentiable functions on $[0,1]$. [closed]
We consider the set
$$\left\{u \in C^2([0,1]) : u''(0)=\alpha_0 u'(0)+\beta_0 u(0)+\gamma_1 u(1), \quad u''(1)=\alpha_1 u'(1)+\beta_1 u(1)+\gamma_0 u(0) \right\}$$
where $\alpha_i,\beta_i,\gamma_i$ ...
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Understanding epsilon-delta proof regarding dense sets
I'm currently self-studying using Spivak's "Calculus" and I wanted to check on my understanding regarding an epsilon-delta proof for dense sets. The first problem was the following:
If $f$ ...
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If $G\in C([0,1])$ and strictly increasing, can we find a sequence $G_n\n C^{\infty}$ with uniformly equicontinuous density?
Let $G:[0,1]\rightarrow [0,1]$ be a strictly increasing and continuous cdf with $G(1)=1$.
I have proven some property for $G\in C^{\infty}([0,1])$ that relies on the continuity of $g(x)=G'(x)$. I hope ...
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Problem with dense set
On ' Set theory with an introduction to real point sets'(Dasgupta, Abhijit ,2014) i found this exercise:
This is interesting because compare the topological (left,1) and order (right,2) definition of ...
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Prove equivalent form of Baire's Category Theorem
I'm trying to prove these two statements of Baire's Category Theorem are equivalent:
Let X complete metric space. A subset of X is meagre if it can be written as the countable union of nowhere dense ...
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Blumberg's theorem
The Blumberg theorem states that for any real function
$𝑓: \mathbb{R}→\mathbb{R}$ there is a dense subset $𝐷$ of $\mathbb{R}$ such that the restriction of $𝑓$ to $D$ is continuous.
In the ...
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Passing from $\mathcal{D}$ to $\mathcal{S}$ using a density argument and extra condition
I saw a proof for the following statement for the space of Schwartz functions $\mathcal{S}$:
$$
\varphi \in \mathcal{S}: \int \varphi = 0 \iff \exists \Phi\in \mathcal{S}: \Phi' = \varphi,
$$
which ...
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Showing a subset is dense in the symmetric tensor product of a Hilbert space
Let $\mathcal{H}$ be a Hilbert space and $\mathcal{H}^{\odot p}$ denote the $p$-fold symmetric tensor product. I want to show that the set
$$U=\text{Span}\{u^{\otimes p} : u\in \mathcal{H}\}$$
is ...
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Questions on proving that every infinite set of real numbers has a countable dense subset
I have been learning answers provided in this question. The answer from Fishbane uses this definition of a dense subset.
A subset $B$ of $A$ is dense in $A$ if for every $\varepsilon > 0$ and any $...
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More on rationally independent subsets of $\mathbb{R}$.
Suppose that $\lambda_{1}, \lambda_{2}, \lambda_{3}\in\mathbb{C}\setminus\{0\}$ and that $\frac{\lambda_2}{\lambda_1}, \frac{\lambda_3}{\lambda_1}\in\mathbb{R}^{+}\setminus\mathbb{Q}$ such that the ...
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How to show that a subset of a normed space isn't dense?
Consider an arbitrary normed vector space $(X, \Vert \cdot \Vert_X)$ and let $Y \subset X$. We say that $Y$ is dense in $X$ if and only if
$$ \forall x \in X, \epsilon > 0, \, \exists y \in Y : \...
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Prove the subspace of $L^p([a,b])$ determined by the step functions on $[a,b]$ is dense in $L^p([a,b])$.
I am reading the proof of the following proposition and got a bit confused about its idea behind the proof:
Proposition$\quad$ Suppose that $[a,b]$ is a closed bounded interval and that $p$ satisfies ...
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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 $...
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Intersection of an uncountable set and an uncountable dense set
Question :
Let $A,B\subseteq\mathbb{R}$ with $A$ being uncountable and $B$ being uncountable and dense.
Must it be true that $A\cap B≠\emptyset?$
Since $(\mathbb{R}\setminus\mathbb{Q})\cap\mathbb{Q}=...
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