I am having a tough time understanding adjoint of a linear map. Consider a linear map between two vector spaces $\, f:V\rightarrow W,$ let us denote $f^*$ to denote its adjoint.
- Accroding to this video https://www.youtube.com/watch?v=SjCs_HyYtSo (around time 5:50) the author explains that adjoint of a linear map is a function from dual of $\,W$ (denoted by $\,W^*$) to the dual of $\,V$ (denoted by $\,V^*$). So this implies $\,f^*:W^*\rightarrow V^*.$
- On the other hand in the pdf http://math.mit.edu/~trasched/18.700.f10/lect17-article.pdf , the adjoint of the linear map is defined as another linear map from $\,W$ to $\,V.$ So this implies $\,f^*:W\rightarrow V.$
Can some body clarify this discrepancy?