example in Rudin in maximum modulus situation Section 5.22 of Rudin says,
"Let $K$ be a compact Hausdorff space, let $H$ be a compact subset of $K$, and let $A$ be a subspace of $C(K)$ such that $1\in A$ and such that $\|f\|_K=\|f\|_H$ for $f\in A.$ Here we used the notation $\|f\|_E=\sup\{|f(x)|:x\in E\}$."
It seems to me that two continuous functions can achieve their maximum magnitude on a subset $H$ but their sum does not. So how is $A$ a subspace?
 A: Perhaps what that passage could have been written more clearly, but I will say what I think may have been intended.
First a simple example: Suppose $K=\{1,2,3\}$ with the discrete topology and $H=\{2,3\}.$
The functions $\begin{array}{l} 1\mapsto 2 \\ 2 \mapsto 3 \\ 3\mapsto 0 \end{array}$ and $\begin{array}{l} 1\mapsto 2 \\ 2 \mapsto 0 \\ 3\mapsto 3 \end{array}$ both attain their maximum value within $H,$ but their sum attains its maximum elsewhere.
But notice that it did not say "Let $A$ be the subspace consisting of functions attaining their maxima in $H,$ but rather, let $A$ be a subspace consisting of functions attaining their maxima in $H.$ In other words, $A$ is not intended to contain all such functions; rather $A$ contains some such functions and $A$ is closed under linear combination.
A: Rudin is just saying suppose $A$ has that property. Example: $A$ could be the linear subspace of $C(\overline{\mathbb D})$ consisting of those functions in $C(\overline{\mathbb D})$ that are harmonic in $\mathbb D.$ In fact, that is exactly what he has in mind in this section called "An abstract approach to the Poisson kernel". 
