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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!!

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2 Answers 2

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“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)

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  1. 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.
  2. 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

  3. You might also want to look at LINK.

  4. An article about CGT in various categories is also interesting and can be found Here.
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