# when defining continuous operators on distributions, why does the adjoint operator need to be continuous?

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...

## 1 Answer

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.

• ahh you're totally right! and this is page 284 of folland's "real analysis: modern techniques and their applications" – sweetpotato May 11 '20 at 11:28
• @sgtswordfish. I need to check my copy when I get home later today. – md2perpe May 11 '20 at 11:48
• I cannot find that comment in my copy of the book, which must be the first edition. (Image and PDF on Scribd) – md2perpe May 11 '20 at 19:25
• I sent an email to Prof. Folland asking him if we miss something, if we misunderstand what he writes, or if this is an actual error in the book. – md2perpe May 11 '20 at 21:35
• Gerald Folland has now answered my question: "You're right: the place where the continuity of T' is really needed is in guaranteeing that the formula <TF, phi> = <F, T'phi> defines TF as a distribution." – md2perpe May 12 '20 at 18:47