I am reading the fifth chapter (on Dual Spaces) from David Luenberger's Optimization by Vector Space Methods. In Section 5.10, the author has defined weak continuity for possibly nonlinear functionals on a normed space $X$, and has hinted at an analogous definition for weak$^\ast$ continuity for (possibly nonlinear?) functionals on the normed dual $X^\ast$, as follows.
I am unable to confidently write down the latter definition however, and would appreciate some help. I am looking a definition that does not (explicitly) use the concept of weak$^\ast$ topology, since the book has not defined that concept as yet.
Here is my attempt:
Suppose $z$ is a functional, not necessarily linear, defined on the normed dual $X^\ast$ of a normed space $X$. Suppose $y_0$ is any fixed element in $X^\ast$. If, to each $\epsilon > 0$ there correspond a $\delta > 0$ and a finite collection $\{x_1, \cdots, x_n\}$ in $X$ such that $$\big\vert [y - y_0, x_i] \big\vert< \delta \text{ for }i = 1, \cdots, n \implies \big\vert z(y) - z(y_0)\big\vert < \epsilon,$$ then $z$ is said to be weak$^\ast$ continuous at $y_0$.
(Here, the symbol [y, x] represents the scalar generated by the action of a functional $y$ in $X^\ast$ on an vector $x$ in $X$. Also, the symbol $z(y)$ represents the scalar generated by the action of the functional $z$ on an element $y$ in $X^\ast$.)
Looking forward. Thanks.
To define weak* convergence on the dual, we define the function $f_x : X^*\rightarrow \mathbb{C}$ by $f_x(a) = [a, x] = \langle a, x\rangle$ notice this is valid as $a$ is a functional so $a(x)$ is a complex number. Now a sequence $(a_n)$ converges if $\forall x\in\ X, \lim f_x(a_n) = f_x(\lim a_n)$. Now you can define weak* continuity of a function on $X^*$ as simply functions which are continuous in this topology.
tl;dr The definition you have written is right.