Editing Stone–Čech compactification (section)

=== An application: the dual space of the space of bounded sequences of reals ===
The Stone–Čech compactification ''β'''''N''' can be used to characterize <math>\ell^\infty(\mathbf{N})</math> (the [[Banach space]] of all bounded sequences in the scalar [[Field (mathematics)|field]] '''R''' or '''C''', with [[supremum norm]]) and its [[dual space]].

Given a bounded sequence <math>a\in \ell^\infty(\mathbf{N})</math> there exists a [[closed ball]] ''B'' in the scalar field that contains the image of {{mvar|a}}. {{mvar|a}} is then a function from '''N''' to ''B''. Since '''N''' is discrete and ''B'' is compact and Hausdorff, ''a'' is continuous. According to the universal property, there exists a unique extension ''βa'' : ''β'''''N''' → ''B''. This extension does not depend on the ball ''B'' we consider.

We have defined an extension map from the space of bounded scalar valued sequences to the space of continuous functions over ''β'''''N'''.

:<math> \ell^\infty(\mathbf{N}) \to C(\beta \mathbf{N}) </math>

This map is bijective since every function in ''C''(''β'''''N''') must be bounded and can then be restricted to a bounded scalar sequence.

If we further consider both spaces with the sup norm the extension map becomes an [[isometry]]. Indeed, if in the construction above we take the smallest possible ball ''B'', we see that the sup norm of the extended sequence does not grow (although the image of the extended function can be bigger).

Thus, <math>\ell^\infty(\mathbf{N})</math> can be identified with ''C''(''β'''''N'''). This allows us to use the [[Riesz–Markov–Kakutani_representation_theorem|Riesz representation theorem]] and find that the dual space of <math>\ell^\infty(\mathbf{N})</math> can be identified with the space of finite [[Borel measure]]s on ''β'''''N'''.

Finally, it should be noticed that this technique generalizes to the ''L''<sup>∞</sup> space of an arbitrary [[measure space]] ''X''. However, instead of simply considering the space ''βX'' of ultrafilters on ''X'', the right way to generalize this construction is to consider the [[Stone space]] ''Y'' of the measure algebra of ''X'': the spaces ''C''(''Y'') and ''L''<sup>∞</sup>(''X'') are isomorphic as C*-algebras as long as ''X'' satisfies a reasonable finiteness condition (that any set of positive measure contains a subset of finite positive measure).