§ Closed graph theorem
- the graph of a function from a banach space to another banach space is a closed subset iff the function is continuous.
- Formally, given , the set is closed in iff is continuous.
§ Proof: Continuous implies closed
- We must show that every limit point of the graph G is in G.
- Let be a limit point. Since everything in metric spaces is equivalent to the sequential definition, this means that .
- Limits in product spaces are computed pointwise, so
- Thus, from above. Now we calculate:
- where we use the continuity of to push the limit inside.
- Thus, which is an element of .
- So an arbitrary limit point is an element of , and thus G is closed. Qed.
§ Proof: closed implies continuous
- Suppose G as defined above is a closed set. We must show that f is continuous, ie, preserves limits.
- Let be a sequence. We must show that .
- Consider as a sequence in . Let the limit of this sequence be . Since G is closed, in G. By defn of , .
- But since (by defn of ), we have that which proves continuity.