Nonlinear Equations - UFRJ

[SEC. 5.1: FEWSPACES 57
Example 5.3. Let M be an open connected subset of C n . Bergman
space A(M) is the space of holomorphic functions defined in M with
finite L 2 norm. When M is bounded, it contains constant and linear
functions, hence M is clearly a non-degenerate fewspace.
Remark 5.4. Condition 1 holds trivially for any finite dimensional
fewnomial space, and less trivially for subspaces of Bergman space.
(Exercise 5.1). Condition 2 may be obtained by removing points from
M.
To each fewspace F we associate two objects: The reproducing
kernel K(x, y) and a possibly degenerate Kähler form ω on M.
Item (1) in the definition makes V (x) an element of the dual
space F ∗ of F (more precisely, the ‘continuous’ dual space or space
of continuous functionals). Here is a classical result about Hilbert
spaces:
Theorem 5.5 (Riesz-Fréchet). Riesz Let H be a Hilbert space. If
φ ∈ H ∗ , then there is a unique f ∈ H such that
φ(v) = 〈f, v〉 H ∀v ∈ H.
Moreover, ‖f‖ H = ‖φ‖ H ∗
For a proof, see [23] Th.V.5 p.81. Riesz-Fréchet representation
Theorem allows to identify F and F ∗ , whence the Kernel K(x, y) =
(V (x) ∗ )(y). As a function of ȳ, K(x, y) ∈ F for all x.
By construction, for f ∈ F,
f(y) = 〈f(·), K(·, y)〉.
There are two consequences. First of all,
K(y, x) = 〈K(·, x), K(·, y)〉 = 〈K(·, y), K(·, x)〉 = K(x, y)
and in particular, for any fixed y, x ↦→ K(x, y) is an element of F.
Thus, K(x, y) is analytic in x and in ȳ. Moreover, ‖K(x, ·)‖ 2 =
K(x, x).
Secondly, Df(y)ẏ = 〈f(·), DȳK(·, y)¯ẏ〉 and the same holds for
higher derivatives.