$\textbf{Closed graph theorem}$
Let $X,Y$ be Banach spaces and $T$ a linear operator from $X$ into $Y$. Suppose that whenever a sequence $(x_n)$ in $X$ converges to some $x$ in $X$ and $(Tx_n)$ converges to some $y\in Y$, it follows that $y=Tx$. Then $T$ is bounded.
$\textbf{Closed graph theorem}$
Let $X,Y$ be Banach spaces and $T$ a linear operator from $X$ into $Y$. Suppose that whenever a sequence $(x_n)$ in $X$ converges to some $x$ in $X$ and $(Tx_n)$ converges to some $y\in Y$, it follows that $y=Tx$. Then $T$ is bounded.
copied
𝐂𝐥𝐨𝐬𝐞𝐝 𝐠𝐫𝐚𝐩𝐡 𝐭𝐡𝐞𝐨𝐫𝐞𝐦
Let 𝑋,𝑌 be Banach spaces and 𝑇 a linear operator from 𝑋 into 𝑌. Suppose that whenever a sequence (𝑥ₙ) in 𝑋 converges to some 𝑥 in 𝑋 and (𝑇𝑥ₙ) converges to some 𝑦∈𝑌, it follows that 𝑦=𝑇𝑥. Then 𝑇 is bounded.