Direct sum of dual space

Question

Let $E$ be a finite vector space and $L_1$, $L_2$ are subspaces. I want to show that :

if $E=L_1\oplus L_2$ than $E^*=L_1^*\oplus L_2^*$.

I attempted:

$\forall x\in E, x=x_1+x_2$ where $x_i\in L_i$

now if $\phi\in E^*$, we have $\phi(x)= \phi(x_1)+\phi(x_2) $.

here I want to consider a forme $\phi_1$ that take $\phi (x_1)$ as image, and the same for $\phi_2$ but I had some strugle.

Furthermore, I search here for a response but I found just this post, so I' wondering : Do I need other conditions to show this?

Answer 1

Note that with finite-dimensional spaces, there is an isomorphism $E \cong E^*$. Suppose it is over a field $\mathbb{F}$, so $L_1 \cong \mathbb{F}^m$, $L_2 \cong \mathbb{F^n}$ and $E \cong \mathbb{F}^{n+m}$.

Then, $L_1^* \cong L_1 \cong \mathbb{F}^n$ and similar for $L_2$.

Hence, $E^* \cong \mathbb{F}^{n+m} \cong \mathbb{F}^n \oplus \mathbb{F}^m \cong L_1^* \oplus L_2 ^*$