I have come across the following question:
Let $P_n$ denote the vector space of polynomials of a complex variable of degree no more than n and let $z_0,z_1,z_2,....,z_n \in \mathbb{c}$ be distinct points. Show that:
$<p,q> = \sum_{i=0}^{n} \overline{p(z_i)} q(z_i) $
defines an inner product on $P_n$. Further show that for all $p \in P_n$
$ | \frac{1}{\sqrt{n+1}} \sum_{i=0}^{n}p(z_i) | \leq ( \sum_{i=0}^{n} |p(z_i)|^2)^{1/2} $.
I have shown that the inner product satisfies the necessary requirements to be classed as such, but am struggling with the second part of the question. The only thing I can think of is that the inequality seems to be in the form of a Cauchy-Schwarz Inequality, with $\sum_{i=0}^{n}q(z_i) = \frac{1}{\sqrt{n+1}}$ and $\|q(z_i)\| = 1$ for some $q(z_i)$.
I would be very grateful if anyone could tell me if this is a valid approach or point me in the right direction.
Let $q(z)=1$ (constant polynomial). Then, by Cauchy-schwarz:
$|\sum_{i=0}^{n}p(z_i) |=|\sum_{i=0}^{n}p(z_i)\overline{q(z_i)} |=|<q,p>| \le <p,p>^{1/2}*<q,q>^{1/2}=( \sum_{i=0}^{n} |p(z_i)|^2)^{1/2}*\sqrt{n+1}$.