0
$\begingroup$

Let $P_2[x]$ be the vector space of all real polynomial from degree $2$ or less, for all $f,g\in P_2[x]$

Prove: $\langle f,g\rangle=\int_{0}^{\infty}f(x)g(x)e^{-x}dx$ Is An Inner Product Over $P_2[x]$

So the properties of an inner product are:

  1. $\langle v,v \rangle\geq 0$ and $\langle v,v \rangle= 0 \iff v\equiv 0$

  2. $\langle v,u \rangle=\langle v,u \rangle$ In real inner space

  3. $\langle v+u,h \rangle=\langle v,h \rangle+\langle u,h \rangle$

So for start I took a general 2 degree polynomial $ax^2+bx+c$

$\langle ax^2+bx+c,ax^2+bx+c \rangle=\int_{0}^{\infty}(ax^2+bx+c)^2e^{-x}dx$

Can it be solve via integration by parts, Or I could use a property of the inner product to solve it?

$\endgroup$
0
$\begingroup$

You have a nonnegative function under integral. What can you say about this function if the integral is equal to zero? If a polynomial has more roots than its degree, what can you say about the coefficients of this polynomial?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.