Prove that $<C(G), C(G)>$ is dual system with bilinear form $$\int_G \phi(x)\psi(x)dx $$ and $\psi(x),\psi(x)\in C(G)$

so here i was trying bilinear form im not getting how to prove non-degenerate

definition of the non-degenerate bilinear form is in Linear Integral Equations By Raimer Kress


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  • $\begingroup$ what is $G$ here ? $\endgroup$ – Phil-W Mar 17 at 4:37
  • $\begingroup$ @Phil-W,,,measurable set $\endgroup$ – Inverse Problem Mar 17 at 15:11
  • $\begingroup$ I'm guessing it's more than that, since you write C(G), you probably mean continuous functions on G right? So it has to be at least a topological space ? $\endgroup$ – Phil-W Mar 17 at 19:32
  • $\begingroup$ @Phil-W...offcurse G is compact mesuarble and C is continuous fucntions on G $\endgroup$ – Inverse Problem Mar 18 at 7:11
  • $\begingroup$ It would help people to help you if you give all the hypothesis you are considering here, and also the notations, and not let them be forced to guess what you mean. For instance, do you suppose that the measure on $G$ is Radon ? $\endgroup$ – Phil-W Mar 19 at 20:06

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