Can We define Fourier transform (In Natural Sense) on function in $L^p$ for any p using denseness property of Schwartz Space?

Question

I was reading Plancheral isometry theorem.

It uses denseness property of Schwartz space S to induce Fourier to transform on $L^2$

Also, S is dense in $L^p$ for all p

So can we used the same argument to induce Fourier to transform any function in $L^p$

I thought it is not possible due to fact for $L^2 $ we have weak Parseval relation which is not available for any other p

Is my argument is correct?

I have already read Tempered distribution I know we can define Fourier transform there which contain Lp but this in distribution sense Which is not natural definition of function I wanted to extend with original function definition

Any Help Hint in this regard will be appreciated