Functional Analysis

Functional Analysis (lecture notes) DR Ivan F. Wilde Mathematics Department King’s College LONDON 2003 1. Banach Spaces Definition 1.1. A (real) complex normed space is a (real) complex vector space X together with a map : X → R, called the norm and denoted k · k, such that (i) kxk ≥ 0, for all x ∈ X, and kxk = 0 if and only if x = 0. (ii) kαxk = |α|kxk, for all x ∈ X and all α ∈ C (or R). (iii) kx + yk ≤ kxk + kyk, for all x, y ∈ X. Remark 1.2. If in (i) we only require that kxk ≥ 0, for all x ∈ X, then k · k is called a seminorm. Remark 1.3. If X is a normed space with norm k · k, it is readily checked that the formula d(x, y) = kx − yk, for x, y ∈ X, defines a metric d on X. Thus a normed space is naturally a metric space and all metric space concepts are meaningful. For example, convergence of sequences in X means convergence with respect to the above metric. Definition 1.4. A complete normed space is called a Banach space. Thus, a normed space X is a Banach space if every Cauchy sequence in X converges (where X is given the metric space structure as outlined above). One may consider real or complex Banach spaces depending, of course, on whether X is a real or complex linear space. Examples 1.5. 1. If R is equipped with the norm kλk = |λ|, λ ∈ R, then it becomes a real normed space. More generally, for x = (x1 , x2 , . . . , xn ) ∈ Rn define µX ¶1/2 n 2 kxk = |xi | i=1 Then Rn becomes a real Banach space (with the obvious component-wise linear structure). 1 1.2 Functional Analysis — Gently Done Mathematics Department In a similar way, one sees that Cn , equipped with the similar norm, is a (complex) Banach space. 2. Equip C([0, 1]), the linear space of continuous complex-valued functions on the interval [0, 1], with the norm kf k = sup{|f (x)| : x ∈ [0, 1]}. Then C([0, 1]) becomes a Banach space. This norm is call