A commutative Banach algebra whose unit

A commutative Banach algebra whose unit A commutative Banach algebra whose unit ball isn't norm compact.

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The unit ball of C([0,1]) is not compact with respect to the supremum norm, since if p_n(x) = xⁿ, then ||p_n || = 1 and (p_n) has no convergent subsequence.
It's well-known that a normed space Y is finite dimensional iff { y Î Y ; ||y || Ł 1} is compact (cf. [Ker, Theorem 2.5-5]). C([0,1]) is infinite dimensional, hence its unit ball is not compact.

Ref.

[Ker] E. Kreyszig, Introductory functional analysis with applications,John Wiley & Sons, 1978.

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On 22 Feb 2001, 00:14.