Hi everyone. I was just reading Evans' book on PDE, and, at some point, it asked to prove that an holder space is a Banach space, and I tried to do that. I just want to ask you if my proof is correct (if you see dumb errors, just notice also that I study EE, so I'm not much into doing proofs . A banach space is defined if : i) it has a norm; ii)Its cauchy series converge to any point of space; We can start with proving i), by saying that an Holder space has a norm by definition. To prove ii), we can make the statement that in a function u, belonging to a Holder space, |u(x) - u(y)|<= C|x-y|^t , with t defined as an arbitrary exponent and c a costant in R. Then, if a Cauchy sequence is convergent to a point, that means that the distance between any two elements u(x) and u(y) will always be smaller than a small value, let's say, e. so d(u(x), u(y))<e. That means that, if we choose a costant c small enough, and eventually, let's make it tend to 0, we would obtain |u(x) - u(y)| <= c|x - y|^t <e, which proves ii.