When considering a real scalar field with Lagrangian $$\mathcal{L} = - \frac{1}{2}(\partial_{\mu} \phi)(\partial^{\mu} \phi) - \frac{1}{2}m^2\phi^2$$ the equation of motion is the Klein-Gordon equation $(\Box_{x} - m^2) \phi(x)=0$.
In texts on QFTs in curved spacetimes the quantization of the field $\phi$ is performed by summing over Minkowski modes $\{ u_{\mathbf{k}} \}_{\mathbf{k} \in \mathbb{R}^3}$ (the positive frequency modes) and $\{ u^{\ast}_{\mathbf{k}} \}_{\mathbf{k} \in \mathbb{R}^3}$ (the negative frequency modes) which are plane-wave solutions to the KG equation labelled by the three-momentum $\mathbf{k} \in \mathbb{R}^3$: $$ u_{\mathbf{k}}(x) \ = \ u_{\mathbf{k}}(x^0,\mathbf{x}) \ = \ \left( (2\pi)^3 2 \sqrt{ |\mathbf{k}|^2 + m^2 } \right)^{-\frac{1}{2}} e^{ - i \sqrt{ |\mathbf{k}|^2 + m^2 }\ x^0 + i \mathbf{k} \cdot \mathbf{x} } $$ The Minkowski modes have normalizations $\left( (2\pi^3) 2 \sqrt{ |\mathbf{k}|^2 + m^2 } \right)^{-\frac{1}{2}}$ because they are normalized with respect to the Klein-Gordon inner product, defined for any complex-valued functions $f,g$ as $$ \langle f,g\rangle = i \int_{\Sigma} d^{3}\mathbf{x} \ \left[ f^{\ast}(x) \frac{\partial g}{\partial x^0} - \frac{\partial f^{\ast}}{\partial x^0} g(x) \right] $$
Where $\Sigma$ is a 3D hypersurface of constant time $x^0$ (because the function being integrated is a conserved current, it follows that the value of $\langle f,g\rangle$ is independent of the choice of $\Sigma$ used to integrate it). The Minkowski modes are normalized such that: $$ \langle u_{\mathbf{k}}, u_{\mathbf{p}} \rangle = \delta(\mathbf{k} - \mathbf{p}) \\ \langle u_{\mathbf{k}}, u^{\ast}_{\mathbf{p}} \rangle = 0 \\ \langle u^{\ast}_{\mathbf{k}}, u^{\ast}_{\mathbf{p}} \rangle = - \delta(\mathbf{k} - \mathbf{p}) $$
At this point, the texts will often say the Minkowski modes are complete, which is why we then are able to expand the scalar field $\phi$ as $\phi(x) = \sum u_{\mathbf{k}}(x) a_{\mathbf{k}} + u^{\ast}_{\mathbf{k}}(x) a^{\dagger}_{\mathbf{k}}$, and then quantize $a_{\mathbf{k}}$ and $a_{\mathbf{k}}^{\dagger}$, and so on.
My Question: What does it precisely mean for the Minkowski modes to be complete here?
The texts all seem to glaze over this point. I want to say that there should be a completeness relation $\sum_{\mathbf{k}} u^{\ast}_{\mathbf{k}}(x)u_{\mathbf{k}}(y)$ which is proportional to either $\delta^{(3)}(\mathbf{x} - \mathbf{y})$ or maybe $\delta^{(4)}(x - y)$ but this doesn't seem to be true. I am not even sure what is the vector space which is complete here.
EDIT 1: I am working in ordinary rectangular Minkowski coordinates (flat space) with metric $\eta^{\mu\nu} = \mathrm{diag}(-1,+1,+1,+1)$. I am interested in how to construct the modes in this simplest case (the curved spacetime texts then generalize this procedure to arbitrary manifolds)
EDIT 2: I am guessing that the way to understand the completeness here is something along the lines as you would in QM. If $\{ |n\rangle \}_{n=1}^{N}$ is a complete set of states for some $N$-dimensional Hilbert space $\mathcal{H}$, then we have $\sum_{n=1}^N |n\rangle\langle n| = \mathbb{I}_{N\times N}$, which allows for the expansion of an arbitrary state $|\psi\rangle$ as $|\psi\rangle = \sum_{n=1}^N \langle n|\psi\rangle |n\rangle $. The expansion done on the field $\phi(x)$ is exactly $\phi(x) = \int d^3\mathbf{k}\ \big[ u_{\mathbf{k}}(x) a_{\mathbf{k}} + u^{\ast}_{\mathbf{k}}(x) a_{\mathbf{k}}^{\dagger} \big]$, where $a_{\mathbf{k}}=\langle u_{\mathbf{k}}, \phi\rangle$ and $a^{\dagger}_{\mathbf{k}}=\langle u^{\ast}_{\mathbf{k}}, \phi\rangle$.
