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.


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.

  • $\begingroup$ Thanks @Steven. Can you help me more about the completion? $\endgroup$ – All Dec 4 '13 at 21:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.