For some complex numbers $\lambda_1, \lambda_2, ..., \lambda_k$, define $$p_i(\lambda) = \prod\limits_{j=1, j\neq i}^k \frac{\lambda - \lambda_j}{\lambda_i - \lambda_j}$$ We now observe that, for any polynomial $P(\lambda)$ of degree less than or equal to $(k-1)$, we have $$P(\lambda) = \sum\limits_{i=1}^k p_i(\lambda)P(\lambda_i).$$
Text is from 'Functional Analysis', by Bachman.
I can't see why we should have this representation. I get that $p_i(\lambda_j) = \delta_{ij}$ - does this have any relevance?
Let $Q(\lambda) = \sum_{i=1}^k p_i(\lambda) P(\lambda_i)$. If $p_i(\lambda_j) = \delta_{ij}$ then $Q(\lambda_j) = \sum_{i=1}^k p_i(\lambda_j) P(\lambda_i) = P(\lambda_j)$ for any $j$. (Do you see why?) Two polynomials of degree $k-1$ must be equal if they are equal if they agree on $k$ points, so $P(\lambda) = Q(\lambda)$ for all $\lambda$.