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-rw-r--r--pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md6
1 files changed, 4 insertions, 2 deletions
diff --git a/pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md b/pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md
index 47ddd3f..1140e45 100644
--- a/pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md
+++ b/pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md
@@ -14,11 +14,13 @@ Let $X$, $Y$ be normed spaces.
We say that a collection $\mathcal{T}$ of bounded linear operators
from $X$ to $Y$ is
{: .mb-0 }
-- *pointwise bounded* if the set $\braces{\norm{Tx} : T \in \mathcal{T}}$ is bounded for every $x \in X$,
+- *pointwise bounded* if the set $\braces{\norm{Tx} : T \in \mathcal{T}}$ is bounded
+ for every $x \in X$,
- *uniformly bounded* if the set $\braces{\norm{T} : T \in \mathcal{T}}$ is bounded.
{% enddefinition %}
-Clearly, every uniformly bounded collection of operators is pointwise bounded since $\norm{Tx} \le \norm{T} \norm{x}$.
+Clearly, every uniformly bounded collection of operators is pointwise bounded
+since $\norm{Tx} \le \norm{T} \norm{x}$.
The converse is true, if $X$ is complete:
{% theorem * Uniform Boundedness Theorem %}