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By Juan Carlos Cabello Píñar

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Extra resources for Análisis Funcional

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Notamos por BV [a, b] al espacio vectorial de las funciones de variable acotada en [a, b]. La aplicación g −→ V (g; [a, b]) define una seminorma en el espacio vectorial BV [a, b]. Nótese que si g es constante, entonces V (g; [a, b]) = 0. De hecho V (g; [a, b]) = 0 si, y sólo si, g es constante en sus puntos de continuidad (se puede probar que toda función de variación acotada es continua en un subconjnto denso de [a, b]). Podemos considerar el subespacio M = {g ∈ BV [a, b]; V (g; [a, b]) = 0}. Así, tomando el cociente por M obtenemos un espacio de Banach que notaremos por N BV [a, b].

Cociente de espacios normados. 31 (que se obtiene de forma similar a su anteriormente comentada versión numérica) permite deducir que la función νp : Lp [0, 1] → R definida por 1 νp (f ) = |f | p 1 p (f ∈ Lp [0, 1]) 0 es una seminorma. } . Es claro que N es un subespacio vectorial de Lp [0, 1] y el espacio cociente Lp [0, 1]/N, que denotaremos por Lp [0, 1], se convierte en un espacio normado, definiendo, como se indicó en la nota precedente, 1 f +N p = |f | p 1 p , ∀f ∈ Lp [0, 1]. 0 El Teorema de Riesz-Fisher garantiza que Lp [0, 1] es un espacio de Banach.

1. 6 Todo espacio finito dimensional es isomorfo a su dual. Veremos además que la abundancia de elementos del dual sirve incluso para computar la norma de cualquier vector de X. 7 Sea X un espacio normado no trivial y x un elemento de X. Entonces existe f ∈ X ∗ tal que f = 1 y f (x) = x . Como consecuencia obtenemos lo siguiente: 1. X ∗ separa los puntos de X, es decir, dados x, y ∈ X con x = y existe f ∈ X ∗ tal que f (x) = f (y). 2. x = m´ax{|f (x)| : f ∈ X ∗ , f = 1} (x ∈ X). El Corolario anterior pone de manifiesto cierta simetría entre las normas de X y de X : f = sup{|f (x)| : x ∈ X, x = 1} (f ∈ X ∗ ) ∗ x = sup{|f (x)| : f ∈ X ∗ , f = 1} (x ∈ X).

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Análisis Funcional by Juan Carlos Cabello Píñar

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