# Equivalence Relation to define the tensor product of Hilbert spaces

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• victorvmotti
In summary: Ultimately, the equivalence relation for the tensor product space of Hilbert spaces should be defined 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)##In summary, the equivalence relation for defining the tensor product space of Hilbert spaces involves comparing a linear combination of pairs on the right hand side to a single pair on the left hand side. The definition should ultimately include the summation of coefficients on both sides.
victorvmotti
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 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)##

victorvmotti said:
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 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)##

## 1. What is an equivalence relation?

An equivalence relation is a binary relation between two objects that satisfies three properties: reflexivity, symmetry, and transitivity. It is a way of comparing two objects and determining if they are equivalent or equal.

## 2. What is the tensor product of Hilbert spaces?

The tensor product of Hilbert spaces is a mathematical operation that combines two Hilbert spaces to create a new, larger Hilbert space. It is used in functional analysis and quantum mechanics to describe the composite systems of multiple particles or systems.

## 3. How is an equivalence relation used to define the tensor product of Hilbert spaces?

An equivalence relation is used to define the tensor product of Hilbert spaces by ensuring that the resulting space is well-defined and satisfies certain properties. By using an equivalence relation, we can ensure that the tensor product is a unique and valid mathematical operation.

## 4. What are the properties of an equivalence relation used in defining the tensor product of Hilbert spaces?

The three properties of an equivalence relation that are used in defining the tensor product of Hilbert spaces are reflexivity, symmetry, and transitivity. These properties ensure that the resulting space is well-defined and satisfies certain algebraic and geometric properties.

## 5. Why is the tensor product of Hilbert spaces important in science?

The tensor product of Hilbert spaces is important in science because it provides a way to mathematically describe composite systems of multiple particles or systems. It is used in quantum mechanics, functional analysis, and other fields to study complex systems and phenomena. It also has applications in data compression, signal processing, and machine learning.

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