Example 4.11 of Baby Rudin

Question

I cannot understand what's the index of summation in (10) and whats $c_{n_{1}...n_{k}}$ and how (10) has finitely many terms.

Answer 1

It means$$\sum_{n_1=0}^{N_1}\sum_{n_2=0}^{N_2}\ldots\sum_{n_k=0}^{N_k}c_{n_1n_2\ldots n_k}x_1^{\,n_1}x_2^{\,n_2}\ldots x_k^{\,n_k},$$for some natural numbers $N_1,N_2,\ldots,N_k$.

Answer 2

There is no specific summation index in expression (10). It is simply indicating that the polynomial $P(x)$ is the sum of terms of the form $x_1^{n_1} \dots x_k^{n_k}$, with each term being multiplied by some complex coefficient $c_{n_1 \dots n_k}$. The qualification in the following sentence says that the expansion of $P(x)$ must contain only finitely many monomial terms, thus excluding expressions with an unlimited number of terms such as $P(x) = 1 + x_1 + x_1^2 + x_1^3 + \dots$ (which is not continuous and not even defined for $|x_1| \ge 1$). It is a somewhat imprecise use of notation.