1 files changed, 26 insertions, 31 deletions
diff --git a/pages/functional-analysis-basics/compact-operators.md b/pages/functional-analysis-basics/compact-operators.md
index b114c24..92e94ba 100644
--- a/pages/functional-analysis-basics/compact-operators.md
+++ b/pages/functional-analysis-basics/compact-operators.md
@@ -3,42 +3,37 @@ title: Compact Operators
 parent: Functional Analysis Basics
 nav_order: 4
 published: false
-# cspell:words
 ---
 
 # {{ page.title }}
 
-{: .definition-title }
-> Definition (Compact Linear Operator)
-> {: #compact-operator }
->
-> A linear operator $T : X \to Y$,
-> where $X$ and $Y$ are normed spaces,
-> is said to be a  *compact linear operator*,
-> if for every bounded subset $M \subset X$
-> the image $TM$ is relatively compact in $Y$.
+{% definition Compact Linear Operator %}
+A linear operator $T : X \to Y$,
+where $X$ and $Y$ are normed spaces,
+is said to be a *compact linear operator*,
+if for every bounded subset $M \subset X$
+the image $TM$ is relatively compact in $Y$.
+{% enddefinition %}
 
-{: .proposition-title }
-> Proposition (Characterisation of Compactness)
->
-> Let $X$ and $Y$ be normed spaces.
-> A linear operator $T : X \to Y$ is compact if and only if
-> for every bounded sequence $(x_n)$ in $X$
-> the image sequence $(Tx_n)$ in $Y$ has a convergent subseqence.
+{% proposition Characterization of Compactness %}
+Let $X$ and $Y$ be normed spaces.
+A linear operator $T : X \to Y$ is compact if and only if
+for every bounded sequence $(x_n)$ in $X$
+the image sequence $(Tx_n)$ in $Y$ has a convergent subsequence.
+{% endproposition %}
 
-{: .proposition-title }
-> Every compact linear operator is bounded.
+{% proposition %}
+Every compact linear operator is bounded.
+{% endproposition %}
 
-{: .proposition-title }
-> Proposition (Compactness of Zero and Identity)
->
-> The zero operator on any normed space is compact.
-> The indentity operator on a normed space $X$ is compact if and only if $X$ has finite dimension.
+{% proposition Compactness of Zero and Identity %}
+The zero operator on any normed space is compact.
+The identity operator on a normed space $X$ is compact if and only if $X$ has finite dimension.
+{% endproposition %}
 
-{: .proposition-title }
-> Proposition (The Space of Compact Linear Operators)
->
-> The set $C(X,Y)$ of compact linear operator from a normed space $X$ into a normed space $Y$
-> form a linear subspace of the space $B(X,Y)$ of bounded linear operators from $X$ into $Y$.
-> If $Y$ is a Banach space, then $C(X,Y)$ is a closed linear subspace of the Banach space
-> $B(X,Y)$ and hence itself a Banach space.
+{% proposition The Space of Compact Linear Operators %}
+The set $C(X,Y)$ of compact linear operator from a normed space $X$ into a normed space $Y$
+form a linear subspace of the space $B(X,Y)$ of bounded linear operators from $X$ into $Y$.
+If $Y$ is a Banach space, then $C(X,Y)$ is a closed linear subspace of the Banach space
+$B(X,Y)$ and hence itself a Banach space.
+{% endproposition %}