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I 'd like to know what is the dual space of the space of all the bounded functions on the set $X$, where $X$ can be any set. Also, I don't assume that the function $f$ is measurable relative to any sigma-field. (Thus, underlying sigma-algebra is just the power set $2^X$).

My conjecture is the dual space should be the space of all the finite measures with at most countable support. At least, integral with respect to such measures are bounded linear functional.

I found in Conway's functional analysis book, Chapter VIII, section 2, Exercises 3 and 4 states that if $(X,B,\mu)$ is a a-finite measure space, the maximal ideal space of $L^\infty(X, B,\mu)$ is totally disconnected.

Does this, combined with Gelfand representation, lead to the answer to my question? If yes, how?

egreg
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gejikeiji
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2 Answers2

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Your space is $\ell^\infty(X)$. The space of complex measures with countable support is the dual of $c_0(X)$ (that is, $C_0(X)$ if we give $X$ the discrete topology); the dual of $\ell^\infty(X)$ is much larger.

For example, let $(x_n)$ be any sequence in $X$. You can use Hahn-Banach to show that there is a bounded linear functional $\Lambda$ such that $\Lambda f=\lim f(x_n)$ for all $f$ for which the limit exists.

One way to describe the dual is as the space of finitely additive complex measures on the power set of $X$; note that a finitely additive measure can be a much stranger thing than you think. (For example, try to imagine what finitely additive measure might give the functional described in the previous paragraph.)

That exercise in Conway doesn't seem directly relevant since counting measure on $X$ is not $\sigma$-finite. But it's not hard to show that if $K$ is the maximal ideal space of $\ell^\infty(X)$ then the Gelfand transform is an isometry onto $C(K)$; hence the dual is equivalent to the space of complex Borel measures on $K$.

David C. Ullrich
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One way to describe the dual space of the space of bounded functions on $X$, which I will denote by $B(X)$, makes use of the Stone-Cech compactification.

More precisely, let $\hat X$ be the Stone-Cech compactification of $X$, where we endow $X$ with the discrete topology. Then, we can identify $B(X)$ with $C(\hat X)$ in the following way: every bounded function $f:X\to\mathbb C$, that is, every element of $B(X)$, has its image contained in some closed, and therefore compact, ball $B$ in $\mathbb C$. Moreover, $f$ is continuous with respect to the discrete topology on $X$, therefore it extends uniquely to a continuous function $\hat f$ on $\hat X$ (note that this extension does not depend on $B$).

It is not hard to check that the map $B(X)\to C(\hat X),~f\mapsto \hat f$ is an isometric isomorphism. Consequently, we can identify the dual of $B(X)$ with the dual of $C(\hat X)$ which, by a consequence of th e Riesz-Markov representation theorem, can in turn be identified with the space of complex regular measures on $\hat X$.

This actually isn't a very concrete description since, as far as I know, there is no concrete description of the Stone-Cech compactification of infinite discrete spaces.

Note that a similar description also works for the dual of the space of bounded continuous functions on a topological space (without imposing any restriction on such space).

Diego Parodi
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