Let . Show that and that the weak*-topology on is the same as the product topology on .
Without further comment, I will interchange between and . Also, I will sometimes abbreviate to simply . A basic open neighborhood of the weak*-topology on is
for some , , and . Also, is defined by and .
Let be a basic open neighborhood of in the product topology on . That is, , such that if and
Note that . Let . Then .
Then, let be a basic open neighborhood of in the weak*-topology, that is . Now, , such that , . Let , such that . Let . Choose , such that .
Thus, if , where and , then .