$\textbf{Banach-Steinhaus theorem}$
Let $X$ be a Banach space, $Y$ a normed space and $\mathcal{F}\subset\mathcal{B}(X,Y)$ a nonempty family of bounded linear operators from $X$ to $Y$. If $$\sup_{T\in\mathcal{F}}\lVert Tx\rVert<\infty\qquad\forall x\in X$$
then$$\sup_{T\in\mathcal{F}}\lVert T\rVert<\infty.$$
$\textbf{Banach-Steinhaus theorem}$
Let $X$ be a Banach space, $Y$ a normed space and $\mathcal{F}\subset\mathcal{B}(X,Y)$ a nonempty family of bounded linear operators from $X$ to $Y$. If $$\sup_{T\in\mathcal{F}}\lVert Tx\rVert<\infty\qquad\forall x\in X$$
then$$\sup_{T\in\mathcal{F}}\lVert T\rVert<\infty.$$
copied
𝐁𝐚𝐧𝐚𝐜𝐡-𝐒𝐭𝐞𝐢𝐧𝐡𝐚𝐮𝐬 𝐭𝐡𝐞𝐨𝐫𝐞𝐦
Let 𝑋 be a Banach space, 𝑌 a normed space and ℱ⊂ℬ(𝑋,𝑌) a nonempty family of bounded linear operators from 𝑋 to 𝑌. If
sup𝑇∈ℱ∥𝑇𝑥∥<∞ ∀𝑥∈𝑋
then
sup𝑇∈ℱ∥𝑇∥<∞.