# Question on dense subset of l^p space

• kostas230
In summary: You would need to show that finite functions (i.e. functions that are non-zero at finitely points) taking only rational values are dense.
kostas230
Let $X$ be an infinite set. Consider the set $l^p(X)$, where $1\leq p < +\infty$, of all complex functions that satisfy the inequality
$$\sup \{\sum_{x\in E} |f(x)|^p: E \subset X, \;\; |E|<\aleph_0 \} < +\infty$$.
The function $\| \|_p: l^p(X)\rightarrow \mathbb{[0,+\infty]}$ defined by
$$\| f \|_p = \sup \{ \left( \sum_{x\in E} |f(x)|^p \right)^{1/p}: E \subset X, \; |E|<\aleph_0 \}$$
makes $l^p(X)$ a complete normed vector space.

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

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

If $\epsilon>0$ there exists a finite subset $E$ of $X$ such that
$$\|f\|_p - \left(\sum_{x\in E}|f(x)|^p \right)^{1/p}<\epsilon$$
$$\| f \|_p - \| g_E \|_p <\epsilon.$$Therefore, if $G$ a finite subset of $X$ that contains $E$ then $\| f \|_p - \| g_G \|_p < \epsilon$. What I've been having problem proving is the inequality $\| f - g_G \|_p < \epsilon$. Any ideas on this? Thanks in advance! :)

How would you prove it for ##X = \mathbb{N}##? Can you mimic that proof?

There seems to be at least one serious typo. $\delta_a(a)=0$ Should be =1?

WWGD
mathman said:
There seems to be at least one serious typo. $\delta_a(a)=0$ Should be =1?

Together with unstated conditions:
1)What is ##E##
2) I think the condition ## || ||_p ## should have range in ## [0, \infty) ##. I am not aware of infinity-valued normed spaces.
3) This may be obvious, but I guess your bet is defined on $l^p(X)$, and not on, some other set?

WWGD said:
Together with unstated conditions:
1)What is ##E##

##E## is finite subset of ##X##.

2) I think the condition ## || ||_p ## should have range in ## [0, \infty) ##. I am not aware of infinity-valued normed spaces.

It does since ##E## is finite.

Ah, yes, my bad, I missed the "## |E| < \aleph_0 ##"; I expected a sort of ## |E| < \infty ##; clearly finite sums are bounded/convergent.

probably the easiest way is to show that finite functions (i.e. functions that are non-zero at finitely points) taking only rational values are dense.
Can you prove that?

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## 1. What is a dense subset in l^p space?

A dense subset in l^p space is a subset of the space that contains points that are arbitrarily close to any point in the space. In other words, every point in the space can be approximated by points in the subset.

## 2. How do you determine if a subset is dense in l^p space?

A subset is dense in l^p space if every point in the space is a limit point of the subset. This means that every neighborhood of a point in the space contains a point from the subset.

## 3. Can a subset be dense in one l^p space but not another?

Yes, a subset can be dense in one l^p space but not another. This is because the definition of density depends on the specific metric or norm used in the l^p space. A subset may be dense in one l^p space but not another if the norms or metrics are different.

## 4. How is a dense subset different from a non-dense subset?

A dense subset contains points that are arbitrarily close to any point in the space, while a non-dense subset does not. In other words, a non-dense subset has points that are far from at least one point in the space.

## 5. What are some examples of dense subsets in l^p space?

Some examples of dense subsets in l^p space include the set of rational numbers in the real numbers, the set of continuous functions in the space of integrable functions, and the set of polynomials in the space of differentiable functions.

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