In functional analysis a famous theorem states that: if $X, Y$ are Banach spaces and $T: X \to Y$ is a linear operator, $T$ is continous if and only if the graph $\Gamma_T:={(x,Tx), x \in X}$ is closed in the product topology. Do you know some nice application of this theorem? Thank you!!
$\endgroup$2 Answers
$\begingroup$“Continuity of a linear map $T:\mathscr X\to\mathscr Y$ means that if $x_n \to x$, then $T(x_n) \to T(x)$, whereas closedness means that if $x_n\to x$ and $T(x_n) \to y$ then $y = T(x)$. Thus the significance of the closed graph theorem is that in verifying that $T(x_n) \to T(x)$ when $x_n \to x$, we may assume that $T(x_n)$ converges to something, and we need only to show that the limit is the right thing. This frequently saves a lot of trouble.” (Folland, 1999, p. 163)
$\endgroup$ $\begingroup$- It is always stated at the beginning when one introduces theory of unbounded operators, to justify that a closed linear operator $A \colon Dom(A) \subset X \rightarrow Y$ such that $Dom(A) = X$ is bounded; which in fact is just a reformulation of CGT.
There is an elementary proof of the fact that for any algebraic isomorphism $\pi \colon \mathcal{A} \rightarrow \mathcal{B}$ between standard operator algebras $\mathcal{A} $ and $\mathcal{B}$ over normed spaces $X$ and $Y$, respectively, there is a linear continuous isomorphism $T\colon X \rightarrow Y$ such that $$ \pi(A) = TAT^{-1} \ \ \text{for all} A \in \mathcal{A}.$$ In particular, $\pi$ is continuous. It uses CGT.
The proof can be found here
You might also want to look at LINK.
- An article about CGT in various categories is also interesting and can be found Here.
