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## Main Question or Discussion Point

I'm following this video on how to establish an equivalence relation to define the tensor product space of Hilbert spaces:

##\mathcal{H1} \otimes\mathcal{H2}={T}\big/{\sim}##

The definition for the

##(\sum_{j=1}^{J}c_j\psi_j, \sum_{k=1}^{K}d_k\varphi_k) \sim \sum_{j=1}^J\sum_{k=1}^Kc_jd_k(\psi_j,\varphi_k)##

But is this correct?

A linear combination of pairs on the right hand side is equivalent to

Shouldn't we define the equivalence relation as below so that we have on

##\sum_{i=1}^Ia_i(\sum_{j=1}^{J}c_j\psi_j, \sum_{k=1}^{K}d_k\varphi_k) \sim \sum_{i=1}^I\sum_{j=1}^J\sum_{k=1}^Ka_ic_jd_k(\psi_j,\varphi_k)##

##\mathcal{H1} \otimes\mathcal{H2}={T}\big/{\sim}##

The definition for the

**equivalence relation**is given in the lecture vidoe as##(\sum_{j=1}^{J}c_j\psi_j, \sum_{k=1}^{K}d_k\varphi_k) \sim \sum_{j=1}^J\sum_{k=1}^Kc_jd_k(\psi_j,\varphi_k)##

But is this correct?

A linear combination of pairs on the right hand side is equivalent to

**only one pair**on the left hand side.Shouldn't we define the equivalence relation as below so that we have on

**both sides linear combination of pairs**?##\sum_{i=1}^Ia_i(\sum_{j=1}^{J}c_j\psi_j, \sum_{k=1}^{K}d_k\varphi_k) \sim \sum_{i=1}^I\sum_{j=1}^J\sum_{k=1}^Ka_ic_jd_k(\psi_j,\varphi_k)##