Dense subspace of linear functionals

Question

We know that any real measurable function can be approximated by increasing simple functions. So, integral of real valued measurable function can be written as a limit of integrals of simple functions. We can observe that the integral of simple functions is just a linear combination of projection maps. I was thinking if this procedure could be done for any linear functional. To be precise, my question is as follows :

Let $X$ be a set and let $L$ be the space of all real valued functions and equip it with uniform norm. Can any linear functional on $L$ be written as a limit of linear combination of projection maps?

By projection map $\pi$, I mean for all $f\in L^0$, $\pi(f) = f(x_0)$ for some $x_0 \in X$ — chandu1729, Oct 22 '13 at 18:47
@copper.hat : Yes, you right. We don't need measure. As I was thinking on the lines of integration, I just introduced measure space. So, we can just take $L$ to be space of all real valued functions. — chandu1729, Oct 22 '13 at 18:53

score 2 · Answer 1 · edited Apr 13 '17 at 12:21

Note that you need to require your functions to be bounded for the norm to make sense. So $L=\ell^\infty(X)$.

Regarding your question, let us take $X=\mathbb N$. It is well-known that there are functionals that annihilate all finitely supported functions. Concretely, you take any free ultrafilter $\omega\in\beta\mathbb N\setminus \mathbb N$ and define a functional $$ \varphi:f\mapsto \bar f(\omega), $$ where $\bar f$ is the extension of $f$ to $\beta\mathbb N$. For any $g\in\ell^\infty(\mathbb N)$ with $g(n)=0$, $n\geq m$ for some $m$, we will have $\varphi(g)=\lim_{n\to\infty}g(n)=0$.

Now let $\varphi_0$ be a "linear combinations of projections maps" as you say: $$ \varphi_0(f)=\sum_{j=1}^hc_jf(n_j) $$ for some fixed $n_1,\ldots,n_h\in\mathbb N$ and $c_1,\ldots,c_h\in\mathbb C$. Now define a function $$ f(n)=\begin{cases}0,&\mbox{ if }n\in\{n_1,\ldots,n_h\}\\ 1,&\mbox{ otherwise} \end{cases}. $$ Then $\|f\|=1$, and $$ |\varphi(f)-\varphi_0(f)|=|1=0|=1. $$ In other words, $\|\varphi-\varphi_0\|\geq1$, i.e. the distance from $\varphi$ to any linear combination of projection maps is at least $1$.

*See this answer for some more information on free-ultrafilters in this context.

Dense subspace of linear functionals

1 Answers1