Try to understand the part of proof -- open mapping theorem

Question

Let $\Lambda$ be a bounded linear operation from $X$ to $Y$, where $X$ and $Y$ are Banach space. Let $B_1(X)$ and $B_1(Y)$ be a unit open ball of the spaces $X$ and $Y$, respectively.

The part that I want to understand is if $\forall y \in B_1(Y), \exists x \in X$ such that $\Lambda x = y$ and $\|x\|\le \rho\|y\|$ for some positive $\rho$, then $B_1(Y) \subset \Lambda (\rho B_1(X))$.

Answer 1

The condition we have says that for $y \in B_1(Y)$ there is an $x \in B_\rho(X)$ such that $\Lambda x = y$. Then we simply notice that $B_\rho(X) = \rho B_1(X)$ so that $x \in \rho B_1(X)$ to get that $y = \Lambda x \in \Lambda(\rho B_1(X))$ which gives the desired inclusion.