Thread: Closed linear span of subspace

Let X be l infinity, consisting of complex sequences (bj) converging to 1/2b1. Let Y be the subset of X consisting of the sequences e1, e2,... together with the element y=e1 + 1/2(sum over k from 2 to infinity)ek. Show that the closed linear span coincides with X...

Many thanks, in advance.

Originally Posted by KSM08

Let X be l infinity, consisting of complex sequences (bj) converging to 1/2b1. Let Y be the subset of X consisting of the sequences e1, e2,... together with the element y=e1 + 1/2(sum over k from 2 to infinity)ek. Show that the closed linear span coincides with X...

Many thanks, in advance.

Do you mean that :the set X is the collection of all infinite complex number sequences which converging to the half of the first term ? and Y is a subset of X consisting of the sequences together with the sequence ?

The definition of the convergence in X is not clear, can you write it down here ?