Let V be a Hilbert space and
and
, the corresponding scalar product and norm, respectively.
A linear form (or linear functional)
on
is a function
such that,
![]() |
(B.1) |
A linear form is bounded if there is a constant
such that,
![]() |
(B.2) |
A bilinear form on is a function
, which is linear in each argument separately, i.e., such that, for all
and
,
![]() |
(B.3) |
![]() |
(B.4) |
The bilinear form
is said to be symmetric if,
![]() |
(B.5) |
bounded if there is a constant
such that,
![]() |
(B.6) |
and
if there is a constant
such that,
![]() |
(B.7) |
The set of all bounded linear functionals on is called dual space of
and denoted
.
The norm in
is given by,
![]() |
(B.8) |
Theorem I (Riesz's representation theorem): Let be a Hilbert space with scalar product
.
For each bounded linear functional
on
there is an unique
such that,
![]() |
(B.9) |
Moreover,
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(B.10) |
Theorem II (Lax-Milgram lemma): If the bilinear form
is bounded and
-elliptic in the Hilbert space
, and
is bounded linear form in
, than there exists a unique vector
such that,
and,
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(B.12) |
Theorem III: Assume that
is a symmetric,
-elliptic bilinear form and that
is a bounded linear form on the Hilbert space
.
Than
satisfies (B.11) if and only if,
![]() ![]() |
(B.13) |