β Scribed by Rudolf Rupp
No coin nor oath required. For personal study only.
Let C R (X) denote, as usual, the Banach algebra of all real valued continuous functions on a compact Hausdorff space X endowed with the supremum norm. We present an elementary proof of the following extension result for C R (X):
For a given g β C R (X) with zero set Zg and for the n-tuple (f1, . . . , fn) β C n R (X) without common zeros in Zg the following assertions are equivalent:
(i) The restriction tuple (f1, . . . , fn)|Z g has an extension to (F1, . . . , Fn) β C n R (X) without common zeros in X. (ii) There exists an n-tuple (h1, . . . , hn) β C n R (X) such that the n-tuple (f1 + h1g, . . . , fn + hng) β C n R (X) has no common zeros in X.
π SIMILAR VOLUMES
We give two equivalent analytic continuations of the MinakshisundaramαPleijel Ε½ . zeta function z for a Riemannian symmetric space of the compact type of U r K rank one UrK. First we prove that can be written as Ε½ . function for GrK the noncompact symmetric space dual to UrK , and F z is an Ε½ Ε½ . .