Lemma 8.1.3. Let $E$ be a TVS over $K \in \RC$, then the topology of $E$ is translation-invariant.
Proof. Let $U \subset E$ open and $y \in E$, then $U + y$ is the preimage of $U$ by the map $x \mapsto x - y$. By (TVS1), $U + y$ is open.$\square$
Lemma 8.1.3. Let $E$ be a TVS over $K \in \RC$, then the topology of $E$ is translation-invariant.
Proof. Let $U \subset E$ open and $y \in E$, then $U + y$ is the preimage of $U$ by the map $x \mapsto x - y$. By (TVS1), $U + y$ is open.$\square$