Análisis Funcional by Juan Carlos Cabello Píñar

By Juan Carlos Cabello Píñar

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Puesto que M es un subespacio, es claro que x + M = d(x, M ), la distancia de x a M. Puede comprobarse fácilmente que la definición no depende del representante elegido y que, la aplicación x + M → x + M , de X/M en K, define una norma en X/M, la cual recibe el nombre de norma cociente en X/M. La consideración de la norma cociente es bastante natural ya que genera la topología cociente en X/M que es, por definición, la mayor topología en X/M que hace continua a la proyección canónica π : X → X/M .

El espacio 1 muestra claramente que la separabilidad de un espacio no es heredada necesariamente por su espacio dual. 4 Sea X un espacio normado cuyo dual X ∗ es separable. Entonces X es separable. 5 Sea X un espacio normado reflexivo. Entonces X es separable si, y sólo si, su dual X ∗ lo es. Como consecuencia, obtenemos que 1 no es reflexivo. Si 1 fuese reflexivo, dado que es separable, también sería separable su dual ∞ , el cual sabemos que no lo es. 2. Dual de C([a, b]) Comencemos definiendo el concepto de integral Riemann-Stieljes.

40 Capítulo 2. Teorema de Hahn-Banach El siguiente resultado es un primer precedente de lo que en teoría de dualidad se conoce como el Teorema del bipolar, a tal efecto introducimos la siguiente definición. 10 Si M es un subespacio de un espacio normado X, el conjunto M ◦ = {f ∈ X ∗ : f (x) = 0, ∀x ∈ M } recibe el nombre de anulador o polar de M en X ∗ . Es claro que M ◦ es un subespacio de X ∗ . El siguiente resultado nos permite decidir cuando un subespacio es denso. 11 Sea M un subespacio de un espacio normado X.

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