# States of the Fock space and the Hilbert space

The Fock space is defined as the direct sum of all $$n$$-particle Hilbert spaces. Are Hilbert space vectors also Fock space vectors or are they just isomorphic to Fock space vectors?

• Could you elaborate the difference between this question and this one (a question by OP)?! Mar 10, 2022 at 12:08

Correct me if I am wrong, but I think it can be thought of through a simpler case. What I mean is that your question is the same as asking if $$a \in \mathbb{R}$$ is also an element of $$\mathbb{R}^n = \bigoplus^n_{i=0} \mathbb{R}$$. Obviously no, because $$\mathbb{R}^n$$ is a vector space, which means that if $$a$$ were an element of the n-dimensional space then we would need a way to add a scalar and an n-dimensional vector, which in most cases is not logical (emphasis on in most cases).
However we can represent $$a$$ in a way that belongs to $$\mathbb{R}^n$$, but that we can agree to identify with $$a$$. For example, if $$a = 1$$ we might say that in $$\mathbb{R}^3$$ $$a = (0, 0, 1)$$.