My question is about a comment in Tao's presentation of the Uniform Boundedness Principle, the Open Mapping theorem, and the Closed Graph theorem in his blog post 245B, Notes 9: The Baire category theorem and its Banach space consequences. He says typically the "easy" directions of the theorems are used in practice while the "hard" directions provide metamathematical justification for the approach.
Strictly speaking, these theorems are not used much directly in practice, because one usually works in the reverse direction (i.e. first proving quantitative bounds, and then deriving qualitative corollaries); but the above three theorems help explain why we usually approach qualitative problems in functional analysis via their quantitative counterparts.
Despite being given three examples of this phenomenon, I still don't understand what he means at all. The examples are below, and I added the boldface to highlight what I don't get. Could someone please expand on this, maybe with a simpler example? I'm aware this is not supposed to be a formal logical concept, but I just don't understand what he's saying.
In the discussion following Example 1 (Fourier inversion formula by Uniform Boundedness), Tao writes
This argument only used the “easy” implication of Corollary 1, namely the deduction of 2. from 3. The “hard” implication using the Baire category theorem was not directly utilised. However, from a metamathematical standpoint, that implication is important because it tells us that the above strategy to prove convergence in norm of the Fourier inversion formula on $L^2$ – i.e. to obtain uniform operator norms on the partial sums, and to establish convergence on a dense subclass of “nice” functions – is in some sense the only strategy available to prove such a result.
In Remark 5 following the Open Mapping Principle, Tao writes
The open mapping theorem provides metamathematical justification for the method of a priori estimates for solving linear equations such as $Lu = f$ for a given datum $f \in Y$ and for an unknown $u \in X$, which is of course a familiar problem in linear PDE. The a priori method assumes that $f$ is in some dense class of nice functions (e.g. smooth functions) in which solvability of $Lu=f$ is presumably easy, and then proceeds to obtain the a priori estimate $\|u\|_X \leq C \|f\|_Y$ for some constant $C$. Theorem 3 then assures that $Lu=f$ is solvable for all $f$ in $Y$ (with a similar bound). As before, this implication does not directly use the Baire category theorem, but that theorem helps explain why this method is “not wasteful.”
Finally, following the Closed Graph Theorem Tao writes
In practice, one should think of $Z$ as some sort of “low regularity” space with a weak topology, and $Y$ as a “high regularity” subspace with a stronger topology. Corollary 3 motivates the method of a priori estimates to establish the $Y$-regularity of some linear transform $Tx$ of an arbitrary element $x$ in a Banach space $X$, by first establishing the a priori estimate $\|Tx\|_Y \leq C \|x\|_X$ for a dense subclass of “nice” elements of $X$, and then using the above corollary (and some weak continuity of $T$ in a low regularity space) to conclude. The closed graph theorem provides the metamathematical explanation as to why this approach is at least as powerful as any other approach to proving regularity.