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In the early 1960s, by using techniques from the model theory of
first-order logic, Robinson gave a rigorous formulation and
extension of Leibniz' infinitesimal calculus. Since then, the
methodology has found applications in a wide spectrum of areas in
mathematics, with particular success in the probability theory and
functional analysis. In the latter, fruitful results were produced
with Luxemburg's invention of the nonstandard hull construction.
However, there is still no publication of a coherent and
self-contained treatment of functional analysis using methods from
nonstandard analysis. This publication aims to fill this gap.
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