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.