Question

inner product spaces?

Answer 1

for functions p=p(x) and q=q(x) in the vector space p3, we define an inner product \[<p,q\]=2p(0)q(0)+p(1)q(1)+p(2)(2). Compute \[|\left| 3-2x+3x ^{2} \right||\]

Answer 2

Your notation leaves some things to be desired... So \(p=p(x)\) and \(q=q(x)\) are polynomial is the vector space \(P_3\) of degree \(3\) polynomials? Or degree \(2\) (thus \(P_3\) as in three terms - constant, linear, quadratic)?

The inner product seems to defined by
\[\langle p,q\rangle=2p(0)q(0)+p(1)q(1)+p(2)q(2)\]Is that also right?

Finally, you want to compute \(\|p(x)\|=\|3-2x+3x^2\|\), which you can do quite easily using the property that for and \(p\in P_3\),
\[\langle p,p\rangle^2=\|p\|\]With the given inner product, you have
\[\langle p(x),p(x)\rangle=2p(0)^2+p(1)^2+p(2)^2~~\implies~~\|p(x)\|=\bigg(2p(0)^2+p(1)^2+p(2)^2\bigg)^2\]

Answer 3

ahh yes yes, thank you!