Originally posted on What's new:

Given a function $latex {f: X \rightarrow Y}&fg=000000$ between two sets $latex {X, Y}&fg=000000$, we can form the graph

$latex \displaystyle \Sigma := \{ (x,f(x)): x\in X \},&fg=000000$

which is a subset of the Cartesian product $latex {X \times Y}&fg=000000$.

There are a number of “closed graph theorems” in mathematics which relate the regularity properties of the function $latex {f}&fg=000000$ with the closure properties of the graph $latex {\Sigma}&fg=000000$, assuming some “completeness” properties of the domain $latex {X}&fg=000000$ and range $latex {Y}&fg=000000$. The most famous of these is the closed graph theorem from functional analysis, which I phrase as follows:

Theorem 1 (Closed graph theorem (functional analysis))Let $latex {X, Y}&fg=000000$ be complete normed vector spaces over the reals (i.e. Banach spaces). Then a function $latex {f: X \rightarrow Y}&fg=000000$ is a continuous linear transformation if and only if the graph $latex {\Sigma := \{ (x,f(x)): x \in X…

View original 1,723 more words