  This paper presents three new families of fractional Sobolev spaces and their
accompanying theory in one-dimension. The new construction and theory are based
on a newly developed notion of weak fractional derivatives, which are natural
generalizations of the well-established integer order Sobolev spaces and
theory. In particular, two new families of one-sided fractional Sobolev spaces
are introduced and analyzed, they reveal more insights about another family of
so-called symmetric fractional Sobolev spaces. Many key theorems/properties,
such as density/approximation theorem, extension theorems, one-sided trace
theorem, and various embedding theorems and Sobolev inequalities in those
Sobolev spaces are established. Moreover, a few relationships with existing
fractional Sobolev spaces are also discovered. The results of this paper lay
down a solid theoretical foundation for systematically developing a fractional
calculus of variations theory and a fractional PDE theory as well as their
numerical solutions in subsequent works. This paper is a concise presentation
of the materials of Sections 1, 4 and 5 of reference [7].
