The discussion confirms that pointwise multiplication of continuous functions does not constitute a valid inner product space. The primary reason is that inner products must yield scalar values, while pointwise multiplication results in another function. The preference for using norms defined by integrals arises from their ability to provide a notion of orthogonality, which is absent in simple product definitions. The analogy with ℝ3 highlights the necessity of summation in defining valid inner products, such as the dot product.
PREREQUISITESMathematicians, students of functional analysis, and anyone interested in the properties of continuous functions and inner product spaces.
aaaa202 said: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?