Understanding the proof in Humphreys Lie algebras page 37

Question

I've been reading 'Introduction to Lie algebras and representation theory' of Humphreys. I have two questions.

The first question is about Coroallary 8.2 in page 37. Corollary states that "The restriction of the Killing form to the maximal toral subalgebra $H$ is nondegenerate." Then, the book says that for each $\phi \in H^*$, there exists the unique $t_\phi \in H$ such that $\phi(h)=\kappa(t_\phi, h)$ for all $h\in H$. I think the Cor. 8.2 only gives a reason for the uniqueness of $t_\phi$, not existence. How can we know that every linear functional on $H$ is represented by the Killing form?

The second question is about the proof of the Proposition 8.3.-(a). (This question is just related to the linear algebra.) The book says that "If $\Phi$ fails to span $H^*$, then (by duality) there exits nonzero $h \in H$ such that $\alpha(h)=0$ for all $\alpha \in \Phi$." Here's what I understand.

Reduce $\Phi$ to be linealry independenet, say $\Phi'$. Consider a subset of $H$ which is dual to $\Phi'$. Since $\Phi'$ cannot span $H^*$, this subset is not a basis for $H$. Extend it to be a basis for $H$. For the basis elements $h_i$'s which are added while extending, $\alpha(h_i)=0$ for all $\alpha \in \Phi'$, and hence for all $\alpha \in \Phi$.

Am I correct? Or is there any easier way to understand it?

Any help would be appreciated. Thanks.

Answer 1

You have $\dim H^*=\dim H$. So, since the map$$\begin{array}{ccc}H&\longrightarrow&H^*\\t&\mapsto&\kappa(\cdot,h)\end{array}$$is injective, it is also surjective.

And if $\langle\Phi\rangle\ne H^*$, take $\beta\in H^*\setminus\{0\}$ such that $(\forall\alpha\in\langle\Phi\rangle):\kappa(\alpha,\beta)=0$. Now, take $h\in H$ such that $\kappa(\cdot,h)=\beta$. Then, if $\alpha\in\langle\Phi\rangle$, $\alpha(h)=\kappa(\alpha,\beta)=0$.