  We study biorthogonal sequences with special properties, such as weak or
weak-star convergence to 0, and obtain an extension of the Josefson-Nissenzweig
theorem. This result is applied to embed analytic disks in the fiber over 0 of
the spectrum of H^infinity (B), the algebra of bounded analytic functions on
the unit ball B of an arbitrary infinite dimensional Banach space. Various
other embedding theorems are obtained. For instance, if the Banach space is
superreflexive, then the unit ball of a Hilbert space of uncountable dimension
can be embedded analytically in the fiber over 0 via an embedding which is
uniformly bicontinuous with respect to the Gleason metric.
