Cc(X) is Banach Algebra

Question

Let X be a compact Hausdorff space. Then Cc(X) is a commutative Banach Algebra with infinity norm. I tried by showing Cc(X) a sub algebra of C(X). I maneged to show that Cc(X) is a vector subspace of C(X) but how to show Cc(X) is closed under ring multiplication. Also I have no idea about how to show that Cc(X) is complete.

Answer 1

There is something not quite right with the question itself. When $X$ is compact, then conspicuously every function in $C(X)$ has compact support so that $C(X)=C_c(X)$.

Maybe the OP wanted to assume that $X$ is locally compact? But then $C_c(X)$ fails to be complete. Indeed, pick a sequence of non-negative, norm-one disjointly supported functions $(f_n)_{n=1}^\infty$ (you can do it by Urysohn's lemma for locally compact spaces). Then the sequence $(\sum_{k=1}^n \frac{f_k}{2^k})_{n=1}^\infty$ is Cauchy, but its limit is not in $C_c(X)$.