From 7c66b227a494748e2a546fb85317accd00aebe53 Mon Sep 17 00:00:00 2001 From: Justin Gassner Date: Thu, 15 Feb 2024 05:11:07 +0100 Subject: Update --- .../the-fundamental-four/closed-graph-theorem.md | 7 ++++--- 1 file changed, 4 insertions(+), 3 deletions(-) (limited to 'pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md') diff --git a/pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md b/pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md index f6a9783..e0ec62b 100644 --- a/pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md +++ b/pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md @@ -33,10 +33,11 @@ This shows that $\graph{T}$ is closed. Conversely, suppose that $\graph{T}$ is a closed subspace of $X \times Y$. Note that $X \times Y$ is a Banach space with norm $\norm{(x,y)} = \norm{x} + \norm{y}$. -Therefore $\graph{T}$ is itself as Banach space in the restricted norm $\norm{(x,Tx)} = \norm{x} + \norm{Tx}$. +Therefore, $\graph{T}$ is itself as Banach space in the restricted norm $\norm{(x,Tx)} = \norm{x} + \norm{Tx}$. The canonical projections $\pi_X : \graph{T} \to X$ and $\pi_Y : \graph{T} \to Y$ are bounded. -Clearly, $\pi_X$ is bijective, so its inverse $\pi_X^{-1} : X \to \graph{T}$ is a bounded operator by the +Clearly, $\pi_X$ is bijective, +so its inverse $\pi_X^{-1} : X \to \graph{T}$ is a bounded operator by the [Bounded Inverse Theorem](/pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.html#bounded-inverse-theorem). -Consequently the composition, $\pi_Y \circ \pi_X^{-1} : X \to Y$ is bounded. +Consequently, the composition, $\pi_Y \circ \pi_X^{-1} : X \to Y$ is bounded. To complete the proof, observe that $\pi_Y \circ \pi_X^{-1} = T$. {% endproof %} -- cgit v1.2.3-54-g00ecf