Can we show $(E \oplus \mathbb R)^* \cong E^* \oplus \mathbb R$, where $E$ is a Banach space and $E^*$ is the dual space of $E$? What if $E$ is just a normed space or even a topological space?
To be specific, if $\Lambda$ is a linear functional on $(E \oplus \mathbb R)^*$, can we find a linear functional $\lambda$ on $E$ and a real number $k \in \mathbb R$ so that, for any $x\in E$ and $y\in\mathbb R$, $$\Lambda(x,y) = \lambda(x) + ky$$
Moreover, can we show $(E \oplus F)^* \cong E^* \oplus F^*$ where $E$ and $F$ are Banach spaces?