The Dual of a Subspace and a Quotient Space

In relation to Claim 2.4, show that if $\mathcal{Z} \leq \mathcal{X}$, $\dim \mathcal{Z} < \infty$, and $\mathcal{M} \leq \mathcal{X}$, then $\mathcal{Z} + \mathcal{M}$ is closed.

Show that if $\mathcal{M} \leq \mathcal{X}$ and $\mathcal{M}$ is topologically complemented in $\mathcal{X}$, then $\mathcal{M}^\perp$ is topologically complemented in $\mathcal{X}^*$ and that its complement is weak-star and linearly homeomorphic to $\mathcal{X}^*/\mathcal{M}^\perp$.