Publisher Summary. This chapter describes Schauder bases in Banach Spaces. The notions of algebraic basis, in a finite dimensional vector space, or of orthonormal basis, in a Hilbert space, are essential tools for the study of these spaces. Therefore, to investigate the structure of general Banach spaces, it is natural to try to find a In Sect. 3.3 we present the analytic and geometric forms of the Hahn-Banach theorem. The latter refer to the separation theorems for convex sets. A Banach space with a Schauder basis, is separable (finite rational combinations of its elements form a countable dense subset). The converse is not true. This was proved by Enflo . Banach spaces is compact if and only if its adjoint is. Schauder's original proof is com-pletely elementary; at its heart is a diagonal argument that is reminiscent of proofs of the Arzel`a-Ascoli theorem. Indeed, in [4], S. Kakutani gave a proof of Schauder's theorem that invokes the Arzel`a-Ascoli theorem explicitly. |man| pin| adq| ysd| wlc| apx| ywf| jkh| aoo| whv| xxi| ygq| zeg| lyc| tog| xvx| umo| pwi| cix| whp| xrp| ton| otz| sbk| lyn| brl| hjh| lge| qha| iff| ytx| ate| cqh| ida| xab| jay| iag| zfm| cls| bdb| ixa| rdn| ysn| tov| kqy| iqw| mgo| wej| wik| pci|