2.4 KiB
Representations of functionals
Lemma 1: let
(X, \langle \cdot, \cdot \rangle)
be an inner product space, if
\forall z \in X: \langle x, z \rangle = \langle y, z \rangle \implies x = y,
and if
\forall z \in X: \langle x, z \rangle = 0 \implies x = 0.
??? note "Proof:"
Will be added later.
Lemma 1 will be used in the following theorem.
Theorem 1: for every bounded linear functional
f
on a Hilbert space(X, \langle \cdot, \cdot \rangle)
, there exists az \in X
such that
f(x) = \langle x, z \rangle,
for all
x \in x
, withz
uniquely dependent onf
and\|z\| = \|f\|
.
??? note "Proof:"
Will be added later.
Sequilinear form
Definition 1: let
X
andY
be vector spaces over the fieldF
. A sesquilinear formh
onX \times Y
is an operatorh: X \times Y \to F
satisfying the following conditions
\forall x_{1,2} \in X, y \in Y: h(x_1 + x_2, y) = h(x_1, y) + h(x_2, y)
.\forall x \in X, y_{1,2} \in Y: h(x, y_1 + y_2) = h(x_1, y_1) + h(x_2, y_2)
.\forall x \in X, y \in Y, \alpha \in F: h(\alpha x, y) = \alpha h(x,y)
.\forall x \in X, y \in Y, \beta \in F: h(x, \beta y) = \overline \beta h(x,y)
.
Hence, h
is linear in the first argument and conjugate linear in the second argument. Bilinearity of h
is only true for a real field F
.
Definition 2: let
X
andY
be normed spaces over the fieldF
and leth: X \times Y \to F
be a sesquilinear form, thenh
is a bounded sesquilinear form if
\exists c \in F: |h(x,y)| \leq c |x| |y|,
for all
(x,y) \in X \times Y
and the norm ofh
is given by
|h| = \sup_{\substack{x \in X \backslash {0} \ y \in Y \backslash {0}}} \frac{|h(x,y)|}{|x| |y|} = \sup_{|x|=|y|=1} |h(x,y)|.
For example, the inner product is sesquilinear and bounded.
Theorem 2: let
(X, \langle \cdot, \cdot \rangle_X)
and(Y, \langle \cdot, \cdot \rangle_Y)
be Hilbert spaces over the fieldF
and leth: X \times Y \to F
be a bounded sesquilinear form. Then there exists a bounded linear operatorsT: X \to Y
andS: Y \to X
, such that
h(x,y) = \langle Tx, y \rangle_Y = \langle x, Sy \rangle_X,
for all
(x,y) \in X \times Y
, withT
andS
uniquely determined byh
with norms\|T\| = \|S\| = \|h\|
.
??? note "Proof:"
Will be added later.