ABSTRACT
The first is called the Schwarz inequality and the second is the parallelogram law. The space L2(0), where n is an open subset of Rn, is an example of a Hilbert space in which the inner product of any two functions f and g is defined by
A linear functional T on the Hilbert space iJC is continuous if and only if
IT(<!>JI "M 11<1>11· for some positive constant M. In the dual space i/C' we define the nonn
However the second dual of 'Je, 'J(" = (iJe')'
(5.4)
and the norm of~ E H"'(O) by
(5.5)
Hence
11<1>11~., ~ ~ lla"<i>lll mE N0 (5.6) 1<>1"'"'
generates the Banach spaces H"'.P(O) (see Exercises 5.1 and 5.2). H"'.P(O) is a Hilbert space only when p = 2, and we shall only be concerned with this case. Consequently H"'·2(0) has been abbreviated to H"'(O).
