Update

1 files changed, 4 insertions, 3 deletions

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 %}