Application of the Hahn-Banach Theorem to the space of bounded sequences with a specific sub linear functional $p$ defined on it gives rise to linear functionals which are dominated by $p$ and are extensions of limits of convergent sequences. These are called Banach Limits and were studied by Banach (1932), and their uniqueness is called almost convergence and was characterised by Lonentz (1948).

In the present lecture we will discuss about the absolute analogue of almost convergence which generalizes lp spaces.

The two concepts of variational inequality and complementarily problems are essentially the same concepts which are studied by two different groups of mathematicians: applied mathematics on one hand and operations researchers on the other hand. The proof existence of variational inequality problem uses Hahn-Banach Theorem or Fixed Point Theorem.

In this lecture we will discuss about the existence of solutions of the complementarily problem, under the most general conditions on the operator and the cone.