Any Schwartz or $L^p$ function $g$ can be identified with a linear functional via which way? $T_g(f)=\int gf$ or $T_g(f)=\int g\bar f$ ? I have seen these two different definitions in different textbooks. Are there any essential difference between them? or either way is acceptable?
A related problem lies in the definition of "linear functional as a function": We say that the linear functional $T$ is the function $g$ if $T(f)=\int gf$. This is the only definition I found so far in textbooks. Can we use the other definition: $T$ can be identified with a function $g$ if $T(f)=\int g\bar f$ ?
People from the PDE world usually work with real-valued functions, and they do not care. If complex-valued functions are important to you, then your question becomes meaningful.
To my eyes, the definition with the bar on $f$ leads to the fact that $$ T_g(\alpha f) = \bar{\alpha}T_g (f), $$ so that $T_g$ is not linear in $f$. I guess you can survive this fact, but people prefer to work with linear rather than conjugate-linear operators.
On the other hand, $(f,g) \mapsto T_g(f)$ could be hermitian, and now you need to put a bar somewhere. To summarize, you should try to understand if either definition yields to useful results for your purposes.