A Basic Question about Interpreting Closed Graph Theorem The closed graph theorem on wikipedia is given as follows:

I'm not understanding how this isn't a tautology. Isn't the definition of continuity that $\{x_n\} \rightarrow x \implies \{f(x_n)\} \rightarrow f(x)$?
 A: There are many, many equivalent definitions for continuity. What you mention is one of them, at least when we are talking for first countable topological spaces. The interesting part in the CGT are numbers 4 and 5. Basically the CGT tells you this: start with a linear operator $T:X\to Y$ between Banach spaces and say you want to check whether this is continuous. With the definition you provided, you would have to take a sequence $\{x_n\}\subset X$ that converges to some $x\in X$ (i.e. $x_n\to x$) and under ONLY this assumption, you have to show that $Tx_n\to Tx$.
Basically, number 4 of the CGT tells you "hey, it's okay, you can ALSO assume that $Tx_n$ converges to some $y\in Y$, i.e. you can ALSO assume that $\{Tx_n\}$ is convergent. You only need to show that $y=Tx$."
You may not see why this is a tremendous help, but when applying it you will understand its magnitude. Here are some interesting applications:

*

*Weakly continuous operators are continuous


*Showing that the composition operator is a bounded linear operator
Here is a suggestion: try to prove (1) without using CGT. Can you do it? (well, even if you can, I bet the answer will be 10 times harder than a simple application of the CGT).
