Consider $C[a,b]$ and let $x_1, \dots, x_{n+1}$ be distinct points in $[a,b]$. Show that there are polynomials $p_{i}$ of degree $n$.

Question

Consider $C[a,b]$ and let $x_1, \dots, x_{n+1}$ be distinct points in $[a,b]$.

a) Show that there are polynomials $p_{i}$ of degree $n$ such that $p_{i}(x_{j}) = \delta_{ij}$ for $1\leq i$, $j\leq n+1$.

b) Deduce that the polynomials of degree at most $n$ form an (n+1)-dimensional subspace.

c) Deduce that $C[a,b]$ is infinite dimensional.

Problem: This question was presented to me at the end of a chapter in my textbook about normed vector spaces. All the section consisted of was defining the norm and showing a few examples of different types of norms. Up to this point the text has covered all of the introductory notions one would take up in a first course in real analysis: sequences, series, functions, differentiation, integration (Riemann).

All that being said I know I have to use the notion of Norm, but I have no idea how to even start. If I get solve part a) then I should be able to work my way through the rest.

Answer 1

Hint for a): Consider the polynomial $$ (x-x_2)(x-x_3)\cdots(x-x_{n+1}) $$ It is almost the polynomial $p_1$ as described in the problem. What is wrong with it? How can you fix that? Can you now see how to construct the other $p_i$?