Norm of the dual of the Tensor product of Hilbert spaces

Question

Let $V$ and $W$ be Hilbert spaces, we can define inner product and induced norm on Tensor product of these spaces as: Let $v_1,v_2 \in V$,and $w_1,w_2 \in W$. then $(v_1 \otimes w_1, v_2 \otimes w_2)_{V\otimes W} := (v_1,v_2)_{V}(w_1,w_2)_{W}$. The dual space of $V \otimes W$ is $(V \otimes W)^{*}$, I also want to define an inner product on this space.How can I do that?

How about for the case $V= H^1_0(D)$ and $W=L^2(\Omega)$ where $D \subset \mathbb{R}$ and $\Omega$ is bounded closed subset of $\mathbb{R}$, and $T>0$. Indeed I need the norm of the dual space.

I want to creet Gelfand triple with $V \otimes W$ and it's dual $(V \otimes W)^{*}$, I Can find a tensor product space like $H$ such that: $$V \otimes W \subset H \subset (V \otimes W)^{*}$$ can I think about $H= L^2(D) \otimes L^2(\Omega)$ as a solution?

Thank you.

Answer 1

You have an inner product on $V\otimes W$, so it(s completion) is a Hilbert space; the dual of a Hilbert space is a Hilbert space, by the Riesz representation theorem, which tells you the inner product.