Ejercicios Funcional

´ EJERCICIOS EN ANALISIS FUNCIONAL

´ Indice 1. Espacios Lineales Normados 1.1. Elementary Properties and Examples 1.2. Convexity 1.3. Geometra 2. Linear Operators 2.1. Continuity of Linear Operators 2.2. Convexity and The Hahn Banach Theorems 2.3. Duality 1 1 3 3 4 4 5 5

1. 1.1.

Espacios Lineales Normados

Elementary Properties and Examples. 1. (Limaye) Let BV [a, b] denote the set ofall functions of bounded variation on [a, b], that is, the set of functions f on [a, b for which the total variation
n

V (f ) := sup
P i=1

|f (xi ) − f (xi−1 )|

2.

3. 4. 5.

6.

7. 8. 9.

10.

is finite (the supremum in V (f ) is taken over all finite partitions P of the form x0 = a < x1 < x2 < . . . < xn−1 < xn = b of [a, b]). Show that V (f ) is a normed linear space.(Conway) If · 1 and · 2 are two norms on a linear space X then they are equivalent (i.e. they define the same topology on X) iff there exists positive constants c and C such that c x 1 ≤ x 2 ≤ C x 1 for all x ∈ X . (Conway) If Y is a finite dimensional subspace of a normed linear space X, then show that Y is closed. In a normed linear space X is the closure of the open ball {x ∈ X : x < 1} equal to theclosed ball {x ∈ X : x ≤ 1}. (Conway) Show that the subspace c of l∞ whose elements are all convergent sequences of real(complex) numbers is closed. In turn, show that the subspace c0 of sequences converging to 0 is a closed subspace of c. Show that c0 and c are isomorphic. (Conway) Show that if X is a locally compact topological space, then the space of continuous functions vanishing at infinity C0(X), consisting of all those functions f ∈ C(X) such that for every ε > 0 there exists K compact such that |f (x)| ≤ ε for all x ∈ X \ K, is a closed subspace of Cb (X) having the norm · ∞ . (Conway) Show that if X is a locally compact topological space, then the space Cc (X) of all continuous functions with compact support is dense in C0 (X) (Limaye) Let X be a normed linear space and let Y be asubspace of X. Show that Y = X iff int Y = ∅. (Conway) A subspace M of a normed linear space is a hyperplane (i.e. dim(X/M ) = 1) iff it is the kernel of a linear functional on X. Two linear functionals have the same kernel iff one is a nonzero multiple of the other. (Conway) A hyperplane M of a normed linear space is either closed or dense.
1

Date: 30 de Septiembre 2011.

11. (Conway) If M is aclosed subspace of a normed linear space X and N is a finite dimensional subspace of X, show that M + N is a closed subspace of X. Show, by an example, that the sum of closed subspaces in a normed linear space is not necessarily closed. 12. (Conway) Let M = {x ∈ lp : x(2n) = 0 for all n}, 1 ≤ p ≤ ∞. Show that lp /M is isometrically isomorphic to lp . 13. (Conway) Let n ≥ 1 and let C n [0, 1] bethe collection of functions f : [0, 1] → R such that f is n-times differentiable with continuous derivatives. Define
n

f := f

∞

+
k=1

f (k)

∞.

Show that C n [0, 1] is a Banach space (i.e. a complete normed space).
∞

14. Demuestre que un espacio normado (X, · ) es completo ssi toda serie
∞ i=1

xi , tal que

xi converge como una sucesi´n de n´meros, converge en X. o u
i=1Test de Weierstrass: Como consecuencia tenemos que, dada una serie de funciones
∞

fn (·) en B(S, Y ), S un conjunto y Y un espacio de Banach, si existe una serie
i=1 ∞

n´merica convergente u
∞ i=1 i=1

ci tal que fn

∞

≤ ci para casi todo i, entonces la serie
∞.

fn (·) de funciones converge uniformemente, es decir, en la norma suprema ·

15. Proporcione un ejemplo de unasucesi´n de funciones en C[0, 1] que converge puntualo mente pero no converge uniformemente (es decir, en la norma suprema). 16. Show that the space C[0, 1] given the norm
1

f :=
0

|f (s)| ds

is not complete. 17. Dado un conjunto S y un espacio lineal normado X, demuestre que el espacio (B(S, X), · de funciones acotadas (sobre S con valores en X) es Banach ssi X es Banach. 18. Let S be...