Is there a theory of higher order functional analysis?

i.e. Instead of taking function spaces, taking functional or operator spaces?

What would this subject be called and are there any known results in it?

Perhaps if function space is like an infinite vector space. Then an example of the higher order (linear) function space would be a like an infinite matrix space? And for the non-linear case it would be a space of infinite dimensional tensors of all orders?

  • $\begingroup$ Not trying to sound cheeky, but functionals and operators are functions themselves, so why wouldn't functional analysis also apply to spaces composed of them? $\endgroup$ – ItsJustASeriesBro Jan 3 at 5:29
  • $\begingroup$ Not sure. But functionals are like functions of functions. They would be one order up in type-theory. $\endgroup$ – zooby Jan 3 at 5:33
  • $\begingroup$ Quadratic forms are also studied in Functional Analysis. $\endgroup$ – DisintegratingByParts Jan 3 at 18:50

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