$V$ is a space of polynomials, we have $p=a_0+a_1x+\dots +a_nx^n$ og $q=b_0+b_1x+\dots +b_nx^n$.

I need to show this function is an inner product:

$$\langle p,q\rangle=\sum_{j=0}^n a_j\overline{b}_j$$

Specifically I need to show it satifies the properties of an inner product: positivity, conjugate symmetry, homogeneity and linearity. I can't get past positivity.

  • $\begingroup$ Is the degree of all polynomials in the space $\leq n?$ $\endgroup$
    – voldemort
    Aug 29 '14 at 2:05
  • $\begingroup$ Out of curiosity, what do you mean by homogeneity? Is that an axiom of inner products? $\endgroup$ Aug 29 '14 at 2:10
  • $\begingroup$ @Patrick: $\langle cv,w\rangle=c\langle v,w\rangle$ $\endgroup$
    – Evanq
    Aug 29 '14 at 2:13
  • $\begingroup$ @voldemort, yes. $\endgroup$
    – Evanq
    Aug 29 '14 at 2:14
  • $\begingroup$ @Evanq oh. I've never heard it called that $\endgroup$ Aug 29 '14 at 2:31

Since the space $V$ contains polynomials of degree $\leq n$, we can do a really easy "cheat".

The map taking $a_0+a_1x+\cdots a_nx^n$ to $(a_0,a_1,\cdots,a_n)$ is a vector space isomorphism from $V$ to $\mathbb{C}^n$. Also, the inner product defined here then becomes the standard inner product on $\mathbb{C}^n$. So, you can now deduce that your "inner product" is really an inner product.

In any case, I would recommend you to go through the proof why the standard inner product on $\mathbb{C}^n$ satisfies the properties like positivity, conjugate symmetry, homogeneity and linearity. The same proof shall apply in this case.


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