An unfamiliar inner product

Consider $E=\mathcal{F}(\R)$ the space of real functions $f:\R \rightarrow \R$ .

Let's define :
$\langle f, g\rangle=f g$ Is $\langle,\rangle$ an inner product ?

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0 useful votes

This is a trick question. All the properties of an inner product seem to be satisfied : it is bilinear, symmetric, and seemingly positive definite. However, an inner product is with values in $\R$ , whereas $\langle f,g \rangle$ is a function.