> [!definition] > > Let $X, Y$ be [[Set|sets]] and $f: X \to Y$ be a [[Function|function]]. The **graph** $\Gamma(f)$ of $f$ is > $ > \Gamma(f) = \bracs{(x, f(x)): x \in X} > $ > [!definition] > > Let $\cx, \cy$ be [[Normed Vector Space|normed spaces]] and $T: \cx \to \cy$ be a [[Linear Transformation|linear map]]. $T$ is **closed** if its graph $\Gamma(T) \subset \cx \times \cy$ is [[Closed Set|closed]] subset with respect to the [[Product Norm|product norm]]. > [!theorem] > > Let $\cx, \cy$ be normed spaces and $T: \cx \to \cy$ be a linear map, then $\Gamma(T)$ is a subspace of $\cx \times \cy$. > > *Proof*. Let $\lambda \in \complex$ and $(x_1, Tx_1), (x_2, Tx_2) \in \Gamma(T)$, then > $ > \lambda (x_1, Tx_1) + (x_2, Tx_2) = (\lambda x_1 + x_2, T(\lambda x_1 + x_2)) \in \Gamma(T) > $ > [!theorem] > > Let $\cx, \cy$ be normed spaces and $T \in L(\cx, \cy)$ be a [[Bounded Linear Map|continuous linear map]], then $T$ is closed. > > *Proof*. Let $\seq{(x_n, Tx_n)} \subset \Gamma(T)$ be a convergent sequence, with $x_n \to x$ and $Tx_n \to Tx$ by continuity. Then its limit $(x, Tx)$ is also on the graph. Therefore the graph is closed.