Here is how folland introduces the construction of continuous operators on distributions:
I understand that the adjoint/equality thing is needed so that the constructed operator is an extension of the original operator.
However, I'm confused about the "continuity of $T'$ guaranteeing continuity of $T$." My question:
For continuity of $T$, why do we need continuity or linearity of $T'$?
To me, it seems that continuity of $T$ only requires that the adjoint $T'$ be a (set) map of the space of test functions $C_c^\infty$ into itself, since if $F_j \to F$ as distributions (ie pointwise on test functions, because weak* topology)
$$ \langle TF_j, \phi \rangle := \langle F_j, T' \phi \rangle \to \langle F, T' \phi \rangle =: \langle TF, \phi \rangle. $$
Actually, based on the above, it seems you could just define a continuous operator $T$ on $D'$ if you have such a set map $T'$.
What am I missing here? I haven't actually checked it, but I bet linearity of $T$ requires linearity of $T'$ was linear...but for continuity...
In which book by Folland did you read this?
I agree with you that continuity of $T'$ seems not be needed for continuity of $T$ extended to $\mathcal{D}'.$ However, I think that it is needed for $TF$ to be a distribution, i.e. for $TF$ to be continuous:
Let $\varphi_j \to \varphi$ in $C_c^\infty$ and assume that $T'$ is continuous. Then, $ \langle TF, \varphi_j \rangle = \langle F, T'\varphi_j \rangle \to \langle F, T'\varphi \rangle = \langle TF, \varphi \rangle ,$ i.e. $TF$ is continuous.