can anyone see the equivalence or relation between the following two definitions of hemicontinuity that I encountered:
Assume that $K$ is a closed, convex subset of Banach space $X$. Let $X^{*}$ be the dual space. Then consider the following two definitions of hemicontinuity:
A mapping $T: K \rightarrow X^{*}$ is said to be hemicontinuous, if the function $t \mapsto \langle T(x + t(y-x)),y-x \rangle$ is continuous at $0^{+}$, for all $x,y$ in $K$.
Then there is another definition as follows:
$T: K \rightarrow X^{*}$ is hemicontinuous if and only if the real function $t \mapsto \langle T(x + ty), z \rangle$ is continuous for all $x,y,z \in K$.
Do these two definitions appear to equivalent or related in any sense? Thanks for any assistance.
Both definitions concern restrictions to lines. The intersection of a line with $K$ is a closed line segment. The second definition requires continuity on the whole segment. The first requires one-sided continuity at one point, but it can be any point, and we can pick the side. So, both definitions require continuity on lines.
The only (possibly) real difference is that the second definition involves evaluating linear functionals on any point $z\in K$, which the first one insists on $y-x$, which is the direction vector of the line under consideration. Thus, the second definition implies the first. It is not clear to me whether the first implies the second.