I'm trying to understand Fock spaces by myself and so I got some very basic doubts I'd like to clarify with you. Let $\mathcal{H}$ be a Hilbert space and $\mathcal{F}(\mathcal{H}):= \bigoplus_{n=0}^{\infty}\mathcal{H}^{\otimes n}$ its associated Fock space. Here, $\mathcal{H}^{\otimes n}$ is the $n$-fold tensor product $\mathcal{H}^{\otimes n} := \mathcal{H}\otimes \cdots \otimes \mathcal{H}$. Now, I'm most interested in bosonic/fermionic Fock spaces. Let $\wedge^{n}\mathcal{H}$ to be the subspace of all antisymmetric tensors of $\mathcal{H}^{\otimes n}$. Then, the fermionic Fock space is $\mathcal{F}_{f}(\mathcal{H}) := \bigoplus_{n=0}^{\infty}\wedge^{n}\mathcal{H}$. If all these being said, let me ask:
(1) As far as I understand, the direct sum $\bigoplus_{n=0}^{\infty}\mathcal{H}^{\otimes n}$ simply means the space of all sequences $(x_{0},x_{1},...)$ with all but finitely many nonzero entries, with $x_{n}\in \mathcal{H}^{\otimes n}$. Thus, it seems natural to define an inner product $\langle \cdot, \cdot \rangle$ on $\mathcal{F}(\mathcal{H})$ by setting: $$\langle x, y \rangle := \sum_{n=0}^{\infty}\langle x_{n},y_{n}\rangle_{\mathcal{H}^{\otimes n}}$$ With this inner product, I believe $\mathcal{F}(\mathcal{H})$ becomes a Hilbert space. Is this reasoning correct?
(2) Related to the above question, some texts I know actually define the same notion of inner product I defined above but directly for fermionic and bosonic Fock spaces. This sounds okay to me as well, but it seems more natural to me to define the inner product on $\mathcal{F}_{f}(\mathcal{H})$ and then restric it to each subspace. Are these constructions equivalent?
(3) Some texts define the Fock space as the space of all sequences $x = (x_{0},x_{1},...)$ with $x_{n} \in \mathcal{H}^{\otimes n}$ satisfying $||x||^{2} := \sum_{n=0}^{\infty}||x_{n}||^{2}_{\mathcal{H}^{\otimes n}} < +\infty$. If the reasoning of my first question is correct, it seems that this definition is just a consequence of my inner product on $\mathcal{F}(\mathcal{H})$, right?
As mentioned in one of the comments, you need a construction as in $(3)$ for your space to be complete whereas your construction in $(1)$ is not necessarily complete, so they are not equivalent. However, if you take the Hilbert space completion of the space you defined in (1), this space will be the same as the space defined in (3).
See the discussion here: https://en.wikipedia.org/wiki/Direct_sum_of_modules#Direct_sum_of_Hilbert_spaces. See also https://en.wikipedia.org/wiki/Fock_space#Definition, the fourth paragraph.