ih
AMREIN BOUTET DE MONVEL-BERTHIER GEORGESCU
if the Banach space V is reflexive (for example, any Hilbertian norm, i.e.
any norm induced by a scalar product, is a reflexive norm). It is easily
shown that a reflexive norm is closed if and only if the function || | || | :
: H [0,+°°] obtained by extending || | || | through the formula: |||u||| = ° ° if
u H ^ V , is lower semi-continuous. It will then be lower semi-continuous
even if we put the weak topology on H. We formulate this as a lemma :
Lemma 2.2: Let || | || | be a norm on a linear subspace V of the Hilbert space
H . Assume || | || | is reflexive, stronger than | | | | and closable. Extend
HI• HI to the completion V c H of (P,|| | || | ) by continuity and then put
H I u H I = * for u £ H ^ P. Then, if { u | n 6U } is a sequence in H such
that u - * u H weakly in H, we have :
lllulH S lim inf ||| u HI .
n--*
We recall the notion of interpolation spaces which will be needed in
Section 4. Let (X,Y) be a Friedrichs couple, i.e. Y is a Hilbert space
and X c z Y is a dense linear subspace provided v/ith a new Hilbert structure
such that the inclusion is continuous. By Friedrichs1 theorem, there is
a unique positive self-adjoint operator A in Y with domain D(A) = X and
| | x||x = J]Ax | |
y
for x X. The interpolation space [X,Y] is defined
for each real number 8 G (0,1) as follows :
[X,Y]fl = DU1-6) ; ||x||[X)Y] = | | A^xHy .
Finally let us mention some simple inequalities which will be fre-
quently used without comment. If u,v,u,.,...,u H and v,s1,...,s are
positive numbers, then
(2.1) 2 | ( u , v ) | S v | | u | |
2
i|| v||
2
,
(2.2) | | £ ?
= 1
u j l
2
n ^
= 1
W^W2
,
Previous Page Next Page