Let . Show that and that the weak*-topology on is the same as the product topology on .

*Remarks*

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 .

*Proof*.

.

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 .

Then

Thus, if , where and , then .

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