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The function is like a dot product given is for polynomials $p(x)$ and $q(x)$: $\langle p,q\rangle =\sum^{n}_{i=0}p(x_i)q(x_i).$ First I had to prove this is an inner product for each $P_k$ where $k\leq n$ and it was pretty straightforward to prove community, linearity, and positive definiteness. I now need to prove it is not an inner product for $k>n$ and am pretty sure this has to do with showing one of those properties doesn't apply but I've tested it a bit and they all seem to work. Does anyone have a hint?

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    $\begingroup$ This question is unclear to me. What is $P_k$? Polynomials of degree $k$? What field are you working over? $\endgroup$ Dec 11, 2017 at 0:28
  • $\begingroup$ Yes sorry I didn't know how to do the polynomial symbol but $P_k$ means polynomials with degree $k$ or less. $\endgroup$
    – Radagast
    Dec 11, 2017 at 0:47

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Consider the polynomial $p(x) = a_{n+1}x^{n+1} + \cdots + a_{k}x^{k}$ and its norm with this inner product.

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  • $\begingroup$ Hmm so I can get $\sqrt{<p,p>}$ to violate something like the Cauchy-Swartz inequality? Some requirement for norms? $\endgroup$
    – Radagast
    Dec 11, 2017 at 0:51
  • $\begingroup$ The norm of a vector is $0$ if and only if the vector is the zero vector. $\endgroup$ Dec 11, 2017 at 0:53

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