Suppose that $X$ and $Y$ are Banach spaces, and that $T_X : X \to Y^*$ and $T_Y : Y \to X^*$ are both isometric isomorphisms. Show that if $(T_X x, y) = (x, T_Y y)$ for all $x \in X, y \in Y$, then $X$ is reflexive.
I don't understand the bracket notation used here. Is $(T_X x,y)$ supposed to mean the image of $y$ under the map $T_X x$? If so I don't understand why the second bracket has them the 'wrong way round'. Clarification would be appreciated.
The bracket notation $(x^*,x)$ for $x\in X$, $x^*\in X^*$ is typically used to denote evaluation of $x^*$ at $x$, i.e., $(x^*,x)=x^*(x)$.
As for your misunderstanding, it could be a typo, and supposed to be $(T_X x, y) = (T_Yy, x)$, or the author may be freely changing coordinates in the parentheses. If the second case is true, I would say that this is sloppy, but it (sort of) reflects the symmetry of the operators $T_X$ and $T_Y$.
In any case, the author is trying to say that $(T_Xx)(y)=(T_Yy)(x)$ for all $x\in X$, $y\in Y$.