# Tensor Product on Hilbert Spaces (well definedness)

Let $H_1$, $H_2$,...,$H_n$ be $n$ Hilbert Spaces. For each $\phi_i \in H_i$, Let $$\phi_1 \otimes \phi_2 \otimes... \otimes \phi_n:= \text{Conjugate multilinear form which acts on H_1 \times H_2.. \times H_n}$$ by

$$(\phi_1 \otimes \phi_2 \otimes... \otimes \phi_n ) \langle\psi_1,\psi_2,..,\psi_n\rangle =\langle\phi_1,\psi_1\rangle_{H_1}\langle\phi_2,\psi_2 \rangle_{H_2}...\langle\phi_n,\psi_n\rangle_{H_n}$$

Let $S$ be the set of all finite linear combinations of such Conjugate multilinear forms. We define an innerproduct $(.,.)$ on $S$ by defining $$\langle\phi_1 \otimes \phi_2 \otimes... \otimes \phi_n ,\psi_1 \otimes \psi_2 \otimes..\otimes \psi_n\rangle =\langle\phi_1,\psi_1\rangle_{H_1}\langle\phi_2,\psi_2 \rangle_{H_2}...\langle\phi_n,\psi_n\rangle_{H_n}$$

We can extend it to whole of $S$ by linearity. Now I want to show that this $(.,.)$ is well defined. For this I need to show that $(\lambda,\lambda')$ doesnot depend on the linear combination which is used to express $\lambda$ and $\lambda'$.

Suppose that $\lambda= \sum_{i=1}^{m}c_i (\phi_{i1} \otimes \phi_{i2} \otimes... \otimes \phi_{in})=\sum_{i=1}^{m}c_i' (\phi'_{i1} \otimes \phi'_{i2} \otimes... \otimes \phi'_{in}$) and $\lambda'=\sum_{i=1}^{m}d_i (\psi_{i1} \otimes \psi_{i2} \otimes... \otimes \psi_{in})=\sum_{i=1}^{m}d'_i (\psi'_{i1} \otimes \psi'_{i2} \otimes... \otimes \psi'_{in})$

From here How do I conclude that it is well defined??

I am learning Tensor Products on Hilbert Spaces.

Thanks for the help!!

• I think this question has been answered already on SE. Here's a link. – Oliver Jones Feb 2 '16 at 10:32
• @OliverJones I want to do it the way I have mentioned – tattwamasi amrutam Feb 2 '16 at 10:37

You don't seem to have set up the approach properly. You should start with the tensor product of the $H_i$'s, i.e. $H=H_1\otimes H_2\otimes \cdots \otimes H_n$. This is a quotient vector space. An expression of the form $\phi_1\otimes \phi_2\otimes \cdots \otimes \phi_n$ represents an equivalence class in $H$. In order for the inner product to be well-defined on $H$, you must show that it doesn't depend on the representative for this class. What you've written above doesn't describe the representatives of $\phi_1\otimes \phi_2\otimes \cdots \otimes \phi_n$. You need to review how $H_1\otimes H_2\otimes \cdots \otimes H_n$ is constructed.