On the non-existence of norms for some algebras of functions
Volume 111 / 1994
Studia Mathematica 111 (1994), 97-101 DOI: 10.4064/sm-111-1-97-101
Let C(Ω) be the algebra of all complex-valued continuous functions on a topological space Ω where C(Ω) contains unbounded functions. First it is shown that C(Ω) cannot have a Banach algebra norm. Then it is shown that, for certain Ω, C(Ω) cannot possess an (incomplete) normed algebra norm. In particular, this is so for $Ω = ℝ^n$ where ℝ is the reals.