  Over the decades, Functional Analysis has been enriched and inspired on
account of demands from neighboring fields, within mathematics, harmonic
analysis (wavelets and signal processing), numerical analysis (finite element
methods, discretization), PDEs (diffusion equations, scattering theory),
representation theory; iterated function systems (fractals, Julia sets, chaotic
dynamical systems), ergodic theory, operator algebras, and many more. And
neighboring areas, probability/statistics (for example stochastic processes,
Ito and Malliavin calculus), physics (representation of Lie groups, quantum
field theory), and spectral theory for Schr\"odinger operators.
  We have strived for a more accessible book, and yet aimed squarely at
applications; -- we have been serious about motivation: Rather than beginning
with the four big theorems in Functional Analysis, our point of departure is an
initial choice of topics from applications. And we have aimed for flexibility
of use; acknowledging that students and instructors will invariably have a host
of diverse goals in teaching beginning analysis courses. And students come to
the course with a varied background. Indeed, over the years we found that
students have come to the Functional Analysis sequence from other and different
areas of math, and even from other departments; and so we have presented the
material in a way that minimizes the need for prerequisites. We also found that
well motivated students are easily able to fill in what is needed from measure
theory, or from a facility with the four big theorems of Functional Analysis.
And we found that the approach "learn-by-using" has a comparative advantage.
