> [!definition] > > Let $E, F$ be [[Topological Vector Space|topological vector spaces]] and $T: E \to F$ be a [[Linear Transformation|linear map]], then the following are equivalent: > 1. $T$ is [[Continuity|continuous]]. > 2. $T$ is continuous at $0$. > > The space $L(E, F)$ is the set of all continuous linear maps from $E$ to $F$. > > As seen in [[Initial Topology on Topological Vector Space|initial topology]], continuous linear map preserve the linear structure, the topological structure, and the compatibility between them. > [!theorem] > > Let $T \in L(E, F)$, then $T$ maps [[Bounded Set|bounded sets]] into bounded sets. > > *Proof*. Let $B \subset E$ be bounded and $V \in \cn^o(0)$ with respect to $F$. By continuity, there exists $V_0 \in \cn^o(0)$ with respect to $E$ such that $T(V_0) \subset V$. Since $B$ is bounded, there exists $\lambda \ge 0$ such that $\lambda V_0 \supset B$, so $\lambda V \supset T(B)$, and $T(B)$ is bounded as well. > [!theorem] > > Let $E$ be a [[Bornologic Space|bornologic space]], $F$ be a locally convex topological vector space, and $T: E \to F$ be a [[Linear Transformation|linear map]], then $T$ is continuous if and only if $T$ maps bounded sets to bounded sets. > > *Proof*. Suppose that $T$ maps bounded sets to bounded sets. Let $V \in \cn^o(0)$ be a convex, balanced, and absorbing neighbourhood of $0$ in $F$, then $T^{-1}(V)$ is a convex and balanced. Let $B \subset E$ be bounded, then $T(B)$ is bounded and is absorbed by $V$, so $B$ is absorbed by $T^{-1}(V)$ as well. Therefore $T^{-1}(V)$ is a neighbourhood of $0$, and $T$ is continuous. > [!theorem] > > Let $E$, $F$ be [[Quasi-Norm|quasi-normed]] spaces and $T: E \to F$ be a linear map, then $T$ is continuous if and only if it maps bounded sets to bounded sets. > > *Proof*. Suppose that $T$ maps bounded sets to bounded sets. Let $\seq{x_k} \subset E$ such that $x_k \to 0$, then there exists a sequence $\seq{n_k} \subset \nat$ such that $n_k \to \infty$ but $n_k \norm{x_k} \to 0$. From here, since $\norm{n_kx_k} \le n_k \norm{x_k}$, $n_kx_k \to 0$ as $k \to \infty$ as well. This allows rewriting > $ > \limv{k}Tx_k = \limv{k}n_k^{-1}Tn_kx_k > $ > where $\seq{n_kx_k}$ and $\seq{Tn_kx_k}$ are bounded. Since $E, F$ are both quasi-normed, there exists $R \ge 0$ such that $\seq{Tn_kx_k} \subset B(0, R)$. Therefore > $ > \limv{k}\norm{Tx_k} \le \limv{k}n_k^{-1}R = 0 > $ > and $T$ is continuous at $0$. > [!theorem] > > Let $(E, \seqi{[\cdot]})$ and $(F, \{[\cdot]_j\}_{j \in J}))$ be [[Locally Convex Topological Vector Space|locally convex topological vector spaces]] and $T: E \to F$ be a linear map, then the following are equivalent: > 1. $T$ is continuous. > 2. For every $j \in J$, there exists a continuous [[Seminorm|seminorm]] $[\cdot]$ on $E$ such that $[Tx]_j \le [x]$ for all $x \in E$. > 3. For every $j \in J$, there exists $\seqf{i_k} \subset I$ and $C \ge 0$ such that $[Tx]_j \le C\sum_{k = 1}^n [x]_{i_k}$ for all $x \in E$.