  This paper introduces a more restrictive notion of feasibility of functionals
on Baire space than the established one from second-order complexity theory.
Thereby making it possible to consider functions on the natural numbers as
running times of oracle Turing machines and avoiding second-order polynomials,
which are notoriously difficult to handle. Furthermore, all machines that
witness this stronger kind of feasibility can be clocked and the different
traditions of treating partial operators from computable analysis and
second-order complexity theory are equated in a precise sense. The new notion
is named "strong polynomial-time computability", and proven to be a strictly
stronger requirement than polynomial-time computability. It is proven that
within the framework for complexity of operators from analysis introduced by
Kawamura and Cook the classes of strongly polynomial-time computable operators
and polynomial-time computable operators coincide.
