7
$\begingroup$

I am trying to understand the notion of density in the context of Banach spaces. The density of a topological space is the least cardinality of a dense subset. Thus, a separable Banach space has density $\omega$.

What is the density of $l_\infty$? What about density of $C([0,1])$? Can one build a Banach space of arbitrary density? Is there a good reference for this topic?

$\endgroup$
  • 2
    $\begingroup$ The density character of $\ell_\infty$ is that of the continuum: it is not separable (eg, there are uncountably many 0-1 sequences), but $\mathbb{R}^\mathbb{N}$ has the same cardinality as $\mathbb{R}$, and $\ell_\infty$ is a subset of this. For C[0,1] the classical Weierstrass approximation theorem says that any function can be approximated by polynomials, which can (in turn) be approximated by polynomials with rational coefficients. To construct a Banach space of arbitrary density consider the functions on some set $K$. This has density character at least that of the cardinality of $K$. $\endgroup$ – James Kilbane Sep 25 '16 at 8:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.