Control 1 - CVV (2004) - Jofre.pdf - Departamento de...

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Departamento de Ingenier´ ıa Matem´atica Facultad de Ciencias F´ ısicas y Matem´aticas UNIVERSIDAD DE CHILE CONTROL 1 MA22A 2004 Prof. Alejandro Jofr´ e Aux. Lilian Rocha Tiempo: 3.5 horas. 1. (a) Sean A, B subconjuntos de un espacio vectorial normado E . (i) Pruebe que adh( A B ) adh( A ) adh( B ). De un ejemplo donde haya igualdad y otro en que la inclusi´on sea estricta. (ii) Suponga que A es cerrado y B es compacto. Muestre, usando la noci´on de sucesiones, que el conjunto A + B = { x E | x = a + b con a A, b B } es cerrado. (b) Sea A = [0 , 1] y considere el espacio vectorial E de las funciones f : A tales que f, f 0 son continuas y acotadas. Consideremos la funci´on k·k : E definida por k f k = k f k + k f 0 k . (i) Muestre que k·k es norma en E . (ii) Verifique que k·k no es equivalente a k·k en E . Para ello considere la sucesi´on ( f n ) n definida mediante f n ( x ) = x n n . 2. Considere la sucesi´on de funciones f n : [ - 1 , 1] definida por f n ( x ) = 0 - 1 x ≤ - 1 /n, n ( x + 1 /n ) - 1
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