This question is regarding a definition of Tensor product of Hilbert spaces that I found in Wald's book on QFT in curved space time. Let's first get some notation straight.

Let $(V,+,*)$ denote a set $V$, together with $+$ and $*$ being the addition and multiplication maps on $V$ that satisfy the vector space axioms. We define the complex conjugate multiplication ${\overline *}:{\mathbb C} \times V \to V$ as

$$

c {\overline *} \Psi = {\overline c} * \Psi,~~\forall~~\Psi \in V

$$

The vector space formed by $(V,+,{\overline *})$ is called the complex conjugate vector space and is denoted by ${\overline V}$.

Given two Hilbert spaces ${\cal H}_1$ and ${\cal H}_2$ and a bounded linear map $A: {\cal H}_1 \to {\cal H}_2$, we define the adjoint of this map $A^\dagger: {\cal H}_2 \to {\cal H}_1$ as

$$

\left< \Psi_2, A \Psi_1 \right>_{{\cal H}_2} = \left< A^\dagger \Psi_2 , \Psi_1 \right>_{{\cal H}_1}

$$

where $\left< ~, ~ \right>_{{\cal H}_1}$ is the inner product as defined on ${\cal H}_1$ (similarly for ${\cal H}_2$) and $\Psi_1 \in {\cal H}_1,~\Psi_2 \in {\cal H}_2$. That such map always exists can be proved using the Riesz lemma.

Here the word "bounded" simply means that there exists some $C \in {\mathbb R}$ such that $$\left\| A(\Psi_1) \right\|_{{\cal H}_2} \leq C \left\| \Psi_1 \right\|_{{\cal H}_1}$$

for all $\Psi_1 \in {\cal H}_1$ and where $\left\| ~~ \right\|_{{\cal H}_1}$ is the norm as defined on ${\cal H}_1$ (similarly for ${\cal H}_2$)

Great! Now for the statement. Here it is.

The tensor product, ${\cal H}_1 \otimes {\cal H}_2$, of two Hilbert spaces, ${\cal H}_1$ and ${\cal H}_2$, may be defined as follows. Let $V$ denote the set of linear maps $A: {\overline {\cal H}}_1 \to {\cal H}_2$, which have finite rank, i.e. such that the range of $A$ is a finite dimensional subspace of ${\cal H}_2$. The $V$ has a natural vector space structure. Define the inner product on $V$ by

$$

\left< A, B \right>_V = \text{tr}\left( A^\dagger B \right)

$$

(The right side of the above equation is well defined, since $A^\dagger B: {\overline {\cal H}}_1 \to {\overline {\cal H}}_1$ has a finite rank). We define ${\cal H}_1 \otimes {\cal H}_2$ to be the Hilbert space completion of $V$. It follows that ${\cal H}_1 \otimes {\cal H}_2$ consists of all linear maps $A: {\overline {\cal H}}_1 \to {\cal H}_2$ that satisfy the Hilbert-Schmidt condition $\text{tr}\left( A^\dagger A \right) < \infty$.

My question is

**1. How does this definition of the Tensor product of Hilbert spaces match up with the one we are familiar with when dealing with tensors in General relativity?**

PS – I also have a similar problem with Wald's definition of a Direct Sum of Hilbert spaces. I have decided to put that into a separate question. If you could answer this one, please consider checking out that one too. It can be found here. Thanks!

## Best Answer

Let there be given a (monoidal) category ${\cal C}$, e.g., the category of finite dimensional vector spaces, the category of Hilbert spaces, etc.

In such a category ${\cal C}$, one typically has the isomorphism

$$\tag{1} {\cal H} \otimes {\cal K} \cong {\cal L}({\cal H}^{*}, {\cal K}), $$

where ${\cal H}^{*}$ is a dual object, and ${\cal L}$ is the pertinent space of morphisms ${\cal H}^{*}\to {\cal K}$.

Often textbooks don't provide the

actual definitionof a tensor product, which is nevertheless at least partly explained on Wikipedia, but instead cheat by using the isomorphism (1) as aworking definitionof a tensor product ${\cal H} \otimes {\cal K}$.