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.(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# A question on linearity of functionals

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