Distance between polynomials in euclidean space

Question

In Euclidean space $R[x]_{\leq n}$consisting of polynomials with scalar product i need to find distance between polynomial $f=2$ and subspace of polynomials with zero constant term.

How can i approach to this problem?

Answer 1

If your inner product is defined by $\langle p,q \rangle=\int_0^{1} p(x)q(x)dx$ then the answer is $2$. To see this let $f_n(x)=2n(\frac 1 n -x)$ for $x <\frac 1 n$ and $0$ otherwise. it is easy to see that the distance between $f_n$ and $2$ tends to $2$. Now Weirstrass approximation shows that required distance is also $2$. [Some details: the distance is obviously $\leq 2$ so we only have to produce some sequence in the subspace whose distance to $2$ tends to $2$. By Weirstrass approximation any continuous function vanishing at $0$ can be approximated uniformly by polynomials vanishing at $0$ and I am using this for $f_n$].