The definition of the convolution of two distribution seems too abstract to me and I have trouble understanding it.The book define this as $(F*F_1)(\phi)=F(F_1^{-}*\phi)$,where $F$ and $F_1$ is both distributions defined as $$F(\phi)=\int_{\mathbb R^d}f(x)\phi(x)dx$$while $F_1^{-}$ is the reflected distribution given by $F_1^{-}(\phi)=F_1(\phi^-)$，here $\phi^-(x)=\phi(-x)$.Also,the convolution of a distribution with a test function is defined by $(F*\psi)(\phi)=F(\psi^-*\phi)$.
My question is that by definition the inner part of $$F(F_1^{-}*\phi)\space \text{is}\space (F_1^{-}*\phi)(\eta)=F_1^{-}(\phi^{-}*\eta)\space $$$$\text{but} \space F_1^{-}(\phi^{-}*\eta)=\int f(x)(\int\phi(y+x)\eta(y)dy)dx$$ which is a definite mumerical value $a$.So what does $F(F_1^{-}*\phi)$ exactly mean?It mean $F(a)$?But by definition the domain of definition of $F$ is a function $\phi$ not a number.Could anyone help me out the confusion
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