Is Pointwise Multiplication a Valid Inner Product for Continuous Functions?

[aaaa202]

Is the space of continuous functions with the innerproduct being the usual product an inner product space? And if so, why is it we always want to use the space of functions with the norm defined by an integral and not just a simple product? Is it because this IP gives us no notion of orthogonality?

 

[pwsnafu]

Is the space of continuous functions with the innerproduct being the usual product an inner product space?

Inner products are scalar valued. Pointwise multiplication results in another function.

And if so, why is it we always want to use the space of functions with the norm defined by an integral and not just a simple product? Is it because this IP gives us no notion of orthogonality?

Consider ℝ³ and think about why the dot product has a summation.