Exercise 9.A.6

Proof. Suppose $T_{}$

C$isinvertible.Then,because$T_C$issurjective,forevery$w $\in V$ there exist $u,v\in V$ such that $T_{}$

C(u + iv) = w + i0$.Thismeansthat$

Thus $Tu=w$. Hence $T$ is surjective and therefore invertible.

Conversely, suppose $T$ is invertible. Let $u,v\in V$. By surjectivity of $T$, there exist $û,\widehat{v}\in V$ such that $Tû=u$ and $T\widehat{v}=v$. Thus

C(û + iv^) = Tû + iTv^ = u + iv.

Since $u$ and $v$ were arbitrary, it follows that $T_{}$

C$issurjectiveandthereforeinvertible.\square$