Why is the function mapping the indices of an isonormal Gaussian process to its respective random variables linear?

Question

First time approaching the book The Malliavin Calculus and Related Topics by Nualart and I see

Why do these calculations show that the mapping $h \rightarrow W(h)$ is linear?

Answer 1

If a random variable $X$ satisfies $E[X^2] = 0$, then $X=0$ with probability $1$. The calculation shows that $$W(\lambda h + \mu g) - \lambda W(h) - \mu W(g) = 0$$ or equivalently that $$W(\lambda h + \mu g) = \lambda W(h) + \mu W(g)$$ with probability $1$.