Direct Sums and Tensor Products

Abstract

The direct sum of a countable family of Hilbert spaces $\mathcal{H}_i$, $i \in \mathbb{N}$, over the same field, is defined as the set of all sequences x _i with $x_i \in \mathcal{H}_i$ for all i such that $\sum_{i=1}^{\infty} \left\Vert{x_i}\right\Vert_i^2 < \infty$ where $\left\Vert{\cdot}\right\Vert_i$ denotes the norm in $\mathcal{H}_i$. The vector space structure and the inner product are defined “component-wise” on this space of sequences. The second section gives first the construction of the (algebraic) tensor product of two vector spaces. When these vector spaces are actually Hilbert spaces $\mathcal{H}_i$, $i=1,2$ an inner product $\langle \cdot,\cdot \rangle$ is defined on the algebraic tensor product $\mathcal{H}_1\otimes\mathcal{H}_2$ by linear extension of the rule $\langle x_1\otimes x_2,y_1\otimes y_2 \rangle =\langle x_1,y_1\rangle_1 \langle x_2,y_2\rangle_2$ for all $x_i, y_i \in \mathcal{H}_i$, where $\langle \cdot,\cdot\rangle_i$ denotes the inner product in $\mathcal{H}_i$. The completion of this inner product space is called the tensor product of the given Hilbert spaces. This completion is described explicitly in the case that the given Hilbert spaces are separable. And there is a straight forward extension to the case of more than two factors. As an illustration of the application of direct sums and tensor products in quantum physics the state spaces of a particle with spin and the state space of multiparticle systems are discussed (the Boson Fock space and the Fermion Fock space).