A.4 GLIMPSES OF FUNCTIONAL ANALYSIS
317
orthocomplement of dom(A) with respect to X, and hence a nontrivial
pair {x, 0} ~ A I. ~)
Remark A.11. After (11), the domain of A* is made of all y such that
the linear partial function x --~ (y, Ax)y be continuous on dora(A), with
respect to the metric of X. This can be used as an alternative definition
of A*: first define its domain this way, then define the image A*y as
the Riesz vector of the linear continuous mapping obtained by extending
x --~ (y, Ax)y to the closure of dora(A), i.e., all X, by continuity.
If dora(A) is not dense, we can always consider A as being of type
X ° ~ Y, where X ° is the closure of dora(A) (equipped with the same
scalar product as X, by restriction), and still be able to define an adjoint,
now of type Y --~ X °.
Note that (A±) ± is the closure of A. Therefore, if an operator A has
an adjoint, and if dora(A*) is dense, the closure of A is A**, the adjoint
of its adjoint. Therefore,
Proposition A.2.
Let A : X --~ Y be a linear operator with dense domain. If
dom(A*)
is dense in
Y, A
is closable.
Its closure is then A**. This is how we proved that div was closable, in
Chapter 5: The domain of its adjoint is dense because it includes all
functions q0 ~ C0°°(D). Indeed, the map b
-o YD q)div
b =- YD b. grad q0
• 2 •
is IL-continuous for such a q0, due to the absence of a boundary term.
As we see here, the weak divergence is simply the adjoint of the operator
2 2
grad" C0°°(D) --~ C0°°(D), the closure of which in L (D) x IL (D), in turn, is
a strict
restriction (beware!) of the weak gradient.
The reader is invited to play with these notions, and to l~rove what
follows: The boundary of D being partitioned S = S h u S ~ as in the
main chapters, start from grad and - div, acting on smooth fields, but
restricted to functions which vanish on S h and to fields which vanish on
S b, respectively. Show that their closures (that one may then denote
grad h and- div b) are mutual adjoints. Same thing with rot h and rot b.
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APPENDIX A Mathematical Background
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