notes/docs/mathematics/functional-analysis/inner-product-spaces/representations-of-functionals.md

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 a z \in X such that

f(x) = \langle x, z \rangle,

for all x \in x, with z uniquely dependent on f and \|z\| = \|f\|.

??? note "Proof:"

Will be added later.

Sequilinear form

Definition 1: let X and Y be vector spaces over the field F. A sesquilinear form h on X \times Y is an operator h: X \times Y \to F satisfying the following conditions

1. \forall x_{1,2} \in X, y \in Y: h(x_1 + x_2, y) = h(x_1, y) + h(x_2, y).
2. \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).
3. \forall x \in X, y \in Y, \alpha \in F: h(\alpha x, y) = \alpha h(x,y).
4. \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 and Y be normed spaces over the field F and let h: X \times Y \to F be a sesquilinear form, then h 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 of h 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 field F and let h: X \times Y \to F be a bounded sesquilinear form. Then there exists a bounded linear operators T: X \to Y and S: 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, with T and S uniquely determined by h with norms \|T\| = \|S\| = \|h\|.

??? note "Proof:"

Will be added later.