Linear Algebra - Inner Products, Functions, and Closet Polynomial

Question

This is the question:

Formulate the linear algebra problem of finding the closet poly $p \in span \{1, t^2\}$ to the function $f(t)=e^tcos(t)$ with respect to the L$^2$ inner product: $\lt f,g\gt = \int^1_0 f(x)g(x)dx$ (do not solve!)

My question is:

1. What is L$^2$?
2. How do you take the inner product of two functions?
3. Are they asking for an orthagonal polynomial to f(t)?
4. Lastly, how do you solve this problem?

Answer 1

1. In this case $L^2=\{f : \int_{0}^{1}|f(x)|^2dx<1\}$.
2. By definition $\langle f,g\rangle=\int_{o}^{1}f(x)g(x)dx$.
3. No, they're asking for such a polynomial $p(x)=a+bx^2$ that $\|f-p\|^2=\langle f-p,f-p\rangle$ is smallest possible.
4. Find projection $f$ onto subspace $\text{span}\{1,t^2\}$, for example here you can find an algorithm.