- #1

kostas230

- 96

- 3

[tex]\sup \{\sum_{x\in E} |f(x)|^p: E \subset X, \;\; |E|<\aleph_0 \} < +\infty [/tex].

The function [itex]\| \|_p: l^p(X)\rightarrow \mathbb{[0,+\infty]}[/itex] defined by

[tex]\| f \|_p = \sup \{ \left( \sum_{x\in E} |f(x)|^p \right)^{1/p}: E \subset X, \; |E|<\aleph_0 \}[/tex]

makes [itex]l^p(X)[/itex] a complete normed vector space.

What I'm trying to show is that there exists a dense subset of [itex]l^p(X)[/itex] with cardinality equal to that of [itex]X[/itex]. For every point [itex]a\in X[/itex] we consider the function [itex]\delta_a: X\rightarrow \mathbb{C}[/itex] with [itex]\delta_a(x) = 0[/itex], if [itex]x\neq 0[/itex], and [itex]\delta_a (a) = 0[/itex].

Let [itex]f\in l^p(X)[/itex]. Consider the collection [itex]\mathcal{C}[/itex] of all finite subsets of [itex]X[/itex]. The relation [itex]\subset[/itex] on [itex]\mathcal{C}[/itex] makes this collection a directed set. For every [itex]E\in\mathcal{C}[/itex], let [itex]g_E = \sum_{a\in E} f(a) \delta_a[/itex]. The mapping [itex]E\rightarrow g_E[/itex] constitutes a net, which I'm trying to show that it converges to [itex]f[/itex].

If [itex]\epsilon>0[/itex] there exists a finite subset [itex]E[/itex] of [itex]X[/itex] such that

[tex]\|f\|_p - \left(\sum_{x\in E}|f(x)|^p \right)^{1/p}<\epsilon[/tex]

which in turn leads to

[tex]\| f \|_p - \| g_E \|_p <\epsilon.[/tex]Therefore, if [itex]G[/itex] a finite subset of [itex]X[/itex] that contains [itex]E[/itex] then [itex]\| f \|_p - \| g_G \|_p < \epsilon [/itex]. What I've been having problem proving is the inequality [itex] \| f - g_G \|_p < \epsilon [/itex]. Any ideas on this? Thanks in advance! :)