need to prove that the following is inner product space.
Prove that \(\displaystyle \, \bigg \langle \,p,\, q\, \bigg \rangle\, =\, \int_{-1}^1\, t^2\, p(t)\, q(t)\, dt\, \) is inner product space on \(\displaystyle \, V\, =\, R_{\leq\, 3}\, [x].\)
I just got stuck in proving the positive-definiteness.
If for example I take p that isn't 0, why does the integral has to be positive? what if I take t to be the root of p? don't I get 0?
Prove that \(\displaystyle \, \bigg \langle \,p,\, q\, \bigg \rangle\, =\, \int_{-1}^1\, t^2\, p(t)\, q(t)\, dt\, \) is inner product space on \(\displaystyle \, V\, =\, R_{\leq\, 3}\, [x].\)
I just got stuck in proving the positive-definiteness.
If for example I take p that isn't 0, why does the integral has to be positive? what if I take t to be the root of p? don't I get 0?
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