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

,