GNS construction of a weight In the theory of quantum groups in the operator algebraic setting, one deals with weights (instead of positive linear functionals). 
Definition: A weight is a function $\phi $ : $A^+ \rightarrow [0, \infty]$ such that 
1) $\phi(a + b) = \phi(a) + \phi(b)$ for all $a, b \in A^+$ 
2) $\phi(\lambda a) = \lambda \phi(a)$ for all $a \in A^+$ and all $\lambda \in \mathbb{R}^+$
In the literature it is said that one can make a unique GNS-construction (By considering $\mathcal{N}_\phi = \{a \in A; \phi(a^*a) < \infty \}$) for any weight on a $C^*$-algebra. However, how can one define the "sesquilinear" form $(a| b) = \phi(b^*a)$ with a weight, since it is not linear?

What you do is to define $\phi $ by linearity on $$\mathcal M_\phi=\text {span}\,\{a\in A^+:\ \phi (a)<\infty\}. $$ Then the inequality from your comment shows that $b^*a\in \mathcal M_\phi $ whenever $a,b\in\mathcal N_\phi $.
