This is probably a pretty soft question.
Setup:
Let $H$ be a Hilbert space over $\mathbb{C}$ with inner product $\langle x| y\rangle$ where $y\mapsto \langle x|y\rangle$ is linear and let $H'$ be the set of antilinear functionals $f:H \rightarrow \mathbb{C}$, i.e., $f(cx+y) =\bar{c} f(x)+f(y)$. By Riesz representation, there exists a unique $v\in H$ such that $f(x)=\langle x|v\rangle$ and that the map $v\mapsto f$ is a linear bijective isometry. We can then define a canonical inner product on $H'$ by $\langle g| f \rangle = \langle w| v\rangle$ where $w,v$ are the corresponding vectors to antilinear functional $g,f$ so that $H'$ is also a Hilbert space.
Question:
Are there any major differences between $H'$ and the regular dual space $H^*$ in terms of formulation except for the fact that one is anti-linear? For example, I would assume that the $k$-tensor product $T^k(H')$ is canonically isomorphic to the space of anti-linear maps $AL(H^k, \mathbb{C})$