Is set

.

closed in Banach space ?

a) , b)

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- Apr 15th 2010, 01:08 AMveljkoClosed set in Banach?
Is set

.

closed in Banach space ?

a) , b) - Apr 15th 2010, 02:45 AMEnrique2
a) Is true because is a continuous linear functional on . If we denote by this continuous mapping the space is

.

b) Is false. To prove it just observe that there are dense subspaces in formed by integrable functions and with integral 0. Look at the Haar system Haar wavelet - Wikipedia, the free encyclopedia.

If the set of functions with integral 0 were closed, this would imply that the closure of the span of the Haar system, that is the whole , would be included in , that certainly is not true.