In my studies of Quantum mechanics, the following problem with complex vector spaces has come up, specifically as regards the inner product in such a space. Now in Shankars "Principles of Quantum Mechanics 2nd ed" on page 8, 9, the linearity/antilinearity property is explained in a way such that the First vector in an inner product would have to be conjugated: $ <V|aW+bZ> = a<V|W>+b<V|Z> whereas <aW+bZ|V> = a^*<W|V>+b^*<Z|V>$, where a and b are complex scalars, $^*$ means conjugate and V,W,Z are vector elements. In Herstein's "Topics in Algebra" however, the opposite is assumed (see page 192/193): here, $(aW+bZ) \cdot V = aW\cdot V+bZ\cdot V$ while $V\cdot (aW+bZ) = a^*V\cdot W+b^*V\cdot Z $.
In other words, according to Shankar, superposition in the first vector will let the scalars come out as conjugates, while according to Herstein, superposition in the second vector will let the scalars come out as conjugates.
Can someone explain this contradiction? Does it have something to do with the braket convention? Thanks!
It seems to me that if you define $$ \mathbf v \cdot \mathbf w = \langle \mathbf w, \mathbf v \rangle $$ you find that both equations are consistent. The main point is that you want $$ \langle c\mathbf w, c\mathbf v \rangle = |c|^2 \langle \mathbf w, \mathbf v \rangle $$ and you get that in both definitions.