Separability of a vector space and its dual

Let’s recall that a topological space is separable when it contains a countable dense set. A link between separability and the dual space is following theorem:

Theorem: If the dual $$X^*$$ of a normed vector space $$X$$ is separable, then so is the space $$X$$ itself.

Proof outline: let $${f_n}$$ be a countable dense set in $$X^*$$ unit sphere $$S_*$$. For any $$n \in \mathbb{N}$$ one can find $$x_n$$ in $$X$$ unit ball such that $$f_n(x_n) \ge \frac{1}{2}$$. We claim that the countable set $$F = \mathrm{Span}_{\mathbb{Q}}(x_0,x_1,…)$$ is dense in $$X$$. If not, we would find $$x \in X \setminus \overline{F}$$ and according to Hahn-Banach theorem there would exist a linear functional $$f \in X^*$$ such that $$f_{\overline{F}} = 0$$ and $$\Vert f \Vert=1$$. But then for all $$n \in \mathbb{N}$$, $$\Vert f_n-f \Vert \ge \vert f_n(x_n)-f(x_n)\vert = \vert f(x_n) \vert \ge \frac{1}{2}$$. A contradiction since $${f_n}$$ is supposed to be dense in $$S_*$$.

We prove that the converse is not true, i.e. a dual space can be separable, while the space itself may be separable or not.

Introducing some normed vector spaces

Given a closed interval $$K \subset \mathbb{R}$$ and a set $$A \subset \mathbb{R}$$, we define the $$4$$ following spaces. The first three are endowed with the supremum norm, the last one with the $$\ell^1$$ norm.

• $$\mathcal{C}(K,\mathbb{R})$$, the space of continuous functions from $$K$$ to $$\mathbb{R}$$, is separable as the polynomial functions with coefficients in $$\mathbb{Q}$$ are dense and countable.
• $$\ell^{\infty}(A, \mathbb{R})$$ is the space of real bounded functions defined on $$A$$ with countable support.
• $$c_0(A, \mathbb{R}) \subset \ell^{\infty}(A, \mathbb{R})$$ is the subspace of elements of $$\ell^{\infty}(A)$$ going to $$0$$ at $$\infty$$.
• $$\ell^1(A, \mathbb{R})$$ is the space of summable functions on $$A$$: $$u \in \mathbb{R}^{A}$$ is in $$\ell^1(A, \mathbb{R})$$ iff $$\sum \limits_{a \in A} |u_x| < +\infty$$.

When $$A = \mathbb{N}$$, we find the usual sequence spaces. It should be noted that $$c_0(A, \mathbb{R})$$ and $$\ell^1(A, \mathbb{R})$$ are separable iff $$A$$ is countable (otherwise the subset $$\big\{x \mapsto 1_{\{a\}}(x),\ a \in A \big\}$$ is uncountable, and discrete), and that $$\ell^{\infty}(A, \mathbb{R})$$ is separable iff $$A$$ is finite (otherwise the subset $$\{0,1\}^A$$ is uncountable, and discrete).

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Determinacy of random variables

The question of the determinacy (or uniqueness) in the moment problem consists in finding whether the moments of a real-valued random variable determine uniquely its distribution. If we assume the random variable to be a.s. bounded, uniqueness is a consequence of Weierstrass approximation theorem.

Given the moments, the distribution need not be unique for unbounded random variables. Carleman’s condition states that for two positive random variables $$X, Y$$ with the same finite moments for all orders, if $$\sum\limits_{n \ge 1} \frac{1}{\sqrt[2n]{\mathbb{E}(X^n)}} = +\infty$$, then $$X$$ and $$Y$$ have the same distribution. In this article we describe random variables with different laws but sharing the same moments, on $$\mathbb R_+$$ and $$\mathbb N$$.

Continuous case on $$\mathbb{R}_+$$

In the article a non-zero function orthogonal to all polynomials, we described a function $$f$$ orthogonal to all polynomials in the sense that $\forall k \ge 0,\ \displaystyle{\int_0^{+\infty}} x^k f(x)dx = 0 \tag{O}.$

This function was $$f(u) = \sin\big(u^{\frac{1}{4}}\big)e^{-u^{\frac{1}{4}}}$$. This inspires us to define $$U$$ and $$V$$ with values in $$\mathbb R^+$$ by: $\begin{cases} f_U(u) &= \frac{1}{24}e^{-\sqrt[4]{u}}\\ f_V(u) &= \frac{1}{24}e^{-\sqrt[4]{u}} \big( 1 + \sin(\sqrt[4]{u})\big) \end{cases}$

Both functions are positive. Since $$f$$ is orthogonal to the constant map equal to one and $$\displaystyle{\int_0^{+\infty}} f_U = \displaystyle{\int_0^{+\infty}} f_V = 1$$, they are indeed densities. One can verify that $$U$$ and $$V$$ have moments of all orders and $$\mathbb{E}(U^k) = \mathbb{E}(V^k)$$ for all $$k \in \mathbb N$$ according to orthogonality relation $$(\mathrm O)$$ above.

Discrete case on $$\mathbb N$$

In this section we define two random variables $$X$$ and $$Y$$ with values in $$\mathbb N$$ having the same moments. Let’s take an integer $$q \ge 2$$ and set for all $$n \in \mathbb{N}$$: $\begin{cases} \mathbb{P}(X=q^n) &=e^{-q}q^n \cdot \frac{1}{n!} \\ \mathbb{P}(Y=q^n) &= e^{-q}q^n\left(\frac{1}{n!} + \frac{(-1)^n}{(q-1)(q^2-1)\cdot\cdot\cdot (q^n-1)}\right) \end{cases}$

Both quantities are positive and for any $$k \ge 0$$, $$\mathbb{P}(X=q^n)$$ and $$\mathbb{P}(Y=q^n) = O_{n \to \infty}\left(\frac{1}{q^{kn}}\right)$$. We are going to prove that for all $$k \ge 1$$, $$u_k = \sum \limits_{n=0}^{+\infty} \frac{(-1)^n q^{kn}}{(q-1)(q^2-1)\cdot\cdot\cdot (q^n-1)}$$ is equal to $$0$$.

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