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In using Hahn-Banach in order to define Banach limits we use limsup as sub-additive dominator on the space on convergent sequences and then extend it to all bounded sequences by H-B. But limsup does not assign a real value to each bounded sequence since a bounded may not be convergent. In conclusion, is a function realvalued if it is realvalued for all $x \in X$ or only for which it is defined? The following question arose after reading section 4.2 in Lax Functional Analysis for those who own a copy.

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The functional $\limsup$ is defined on the whole of $\ell^\infty$ (the bounded sequences), you seem to mix up $\limsup$ with $\lim$. Any bounded sequence has accumulation points (as bounded sets are compact), and hence, by completeness of $\mathbf R$, a largest accumulation point, its limes superiror.

The $\limsup$ of a bounded sequence $(x_n)$ is given by $$ \limsup_{n\to\infty}x_n = \lim_{n\to \infty} \sup_{k \ge n} x_k $$ Note that $(\sup_{k\ge n} x_k)_n$ is a bounded, decreasing sequence and hence, its limit exists.

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