Functional Analysis and the Feynman Operator Calculus

Kuelbs–Steadman Spaces for Banach Space-Valued Measures

“…getting the last part of the assertion. Thus, the embeddings in (23) and (24) are dense. , respectively.…”

“…The proof of the case p = ∞ is analogous to that of the case 1 ≤ p < ∞. Therefore, f ∈ KS p [µ], and the embeddings in (23) and (24) [27, §4.4]).…”

“…Kuelbs-Steadman spaces have been the subject of many recent studies (see e.g. [22,23,26] and the references therein). The investigation of such spaces arises from the idea to consider the L 1 spaces as embedded in a larger Hilbert space with smaller norm, and containing in a certain sense the Henstock-Kurzweil integrable functions.…”

“…This condition is satisfied, for instance, when T is a metrizable separable space, not necessarily with a Schauder basis (such spaces exist, see for instance [22]), and µ is a Radon measure. In the literature, some deeply investigated particular cases are when X = R n and µ is the Lebesgue measure, and when X is a Banach space with a Schauder basis (see also [22,23,26]). Since the integral of f with respect to µ is an element of X, in general it is not natural to define an inner product, when it is dealt with norm convergence of the involved integrals.…”

Feynman path formula for the time fractional Schrödinger equation

“…Historically, there are two approaches to study the Feynman path integral, either we take the definition of Feynman path integral as it is presented in [8] and deduce the Schrödinger picture from that, or starting from Schrödinger equation, we try to end up at Feynman path integral (see [9,10,11,12]). In [1], we can find the first approach and see how one can derive the Schrödinger equation from the Feynman path integral.…”