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\centerline{\bf Convergence, Minimax and Monotonicity}
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\centerline{S. Simons}
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\centerline{\bf Abstract}
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In the first lecture, we prove without measure theory a version of
Fatou's lemma for continuous functions on [0,1], using only
that the uniform limit of continuous functions is continuous, and that
every continuous function on [0,1] attains its maximum.   The
proof depends on an eigenvector property of a certain family of
matrices, and can be adapted to prove a result on convergence in a
general Banach space. In the second talk, we show how a generalization of
the Hahn-Banach theorem due to Mazur and Orlicz can be used to prove a
classical minimax theorem, and we also show the connection between
minimax theorems and weak compactness in a Banach space.   In the
remaining three talks, we define monotone and maximal monotone
multifunctions on a Banach space, and show how the minimax theorem can
be used to prove a number of results about these multifunctions.   More
details can be found in the TeX version of this abstract.
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