$||*||_p$ p-Norm for Vector That is Not in $R^n$

Question

My instructor gave us a problem asking to solve the 2-Norm of p also written as$||p||_2$

However, p is in an inner product space $V=C[0,1]$ which means it is a function.

(Specifically, p $=\frac{3}{2}x$)

I'm not sure how to compute the 2-Norm for a function because I thought it was only applicable to vectors in $R^n$?

Answer 1

If $f\in C([0,1])$ then $$ ||f||_2=\Big(\int_0^1|f(x)|^2\;dx\Big)^{\frac{1}{2}}$$ In fact, for each real number $1\leq p<\infty$ there is a corresponding $p$-norm, defined by $$|f||_p=\Big(\int_0^1|f(x)|^p\;dx\Big)^{\frac{1}{p}}$$