Suppose your QFT has a Hilbert space $\mathcal{H}$, and let $\text{End}(\mathcal{H})$ be the set of operators on $\mathcal{H}$. It is often stated that in QFT there is a map $$\mathcal{F}: \text{End}(\mathcal{H}) \rightarrow \mathcal{H}$$ which is straightforward to define in operator language: For any operator $\phi$, define $$\mathcal{F}(\phi) = \phi |0\rangle$$ where $|0\rangle$ is some canonical vacuum state.
However, what's confusing me is the description of this map using the path integral, which goes as follows: If your QFT is defined on a manifold $M$, with non-trivial boundary $\partial M$, then doing a path integral requires some boundary conditions, i.e one has to specify field configurations on $\partial M$, $x|_{\partial M} = x_0$. It is then claimed that this space of boundary field configurations forms a Hilbert space $V$, and the path integral assigns to each operator a linear functional on $V$, a member of $V^*$ that is we have a map $$\text{PI}: \text{End}(\mathcal{H}) \rightarrow V^*.$$ This is defined as follows: Let $x_0 \in V$, and $\phi \in \text{End}(\mathcal{H})$ The linear functional is defined as $$\text{PI}(\phi) (x_0) = \int_{x(\partial M) = x_0} \mathcal{D}x \,\, e^{-S[x]} \, \varphi.$$ Here inside the path integral, $\varphi$ is the classical field corresponding to the operator $\phi$. From this it's clear that $\text{PI}$ assigns a number to a boundary field configuration, however there are several things that are unclear:
What is $V$ precisely? The "space of boundary field configurations" is a bit vague, because if your target space is $X$, then this space consists of all maps from $\partial M$ to $X$, which it isn't even clear is a vector space (how do we add two such maps?).
Why does $V$ (or $V^*$) coincide with the QFT Hilbert space $\mathcal{H}$? For example, assume we have the quantum mechanical system of the harmonic oscillator with $M = [0,\infty)$. Then $\partial M = \{0\}$, and the space of boundary field configurations is simply $\mathbb{R}$, whereas the quantum mechanical Hilbert space of this system is $L^2(\mathbb{R})$.
Why is $\text{PI}(\phi)$ linear in boundary field configurations? This isn't clear from the definition at all.
I have looked at a couple of sources on this and all I can find is a one or two paragraph description of this map, which neglects all the issues I have mentioned above. Can anyone please answer my queries, or suggest a source which discusses them in some detail?