Ricky2357
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Suppose we have a bounded linear functional f defined on L1 (the sequence space of all absolutely summable sequences) and we take the natural (Schauder) basis for L1, that is, the set of sequences (E1,E2,...,En,...) that have 1 in the n th position and everywere else zero. Pick x in L1.
Then x=A1*E1+A2*E2+... , for some scalars An.
Do we need to justify the fact that
f(x)=A1*f(E1)+A2*f(E2)+... ?
In other words, do we need to justify that linearity is still applied even on infinite sums?
Then x=A1*E1+A2*E2+... , for some scalars An.
Do we need to justify the fact that
f(x)=A1*f(E1)+A2*f(E2)+... ?
In other words, do we need to justify that linearity is still applied even on infinite sums?