The term functional is used in at least two different meanings.

One meaning is in the mathematical topic of functional analysis, where one in particular studies linear functionals.
This meaning is not relevant for the discussion at page 299 in Ref. 1.

Another meaning is in the topics of calculus of variations and (classical) field theory. This is the sense that is relevant here.

Since we are only discussing the classical action $S$ and not the full path integral, let us for simplicity forget about quantum aspects, such as, e.g., $\hbar$, Hilbert spaces, expectation values, etc.

Let us for simplicity assume that there is only one field $q$ (which we for semantical reasons will call a position field), and that it lives in $n$ spatial dimensions and one temporal dimension. The field $q$ is then a function $q:\mathbb{R}^{n+1}\to \mathbb{R}$. There is also a velocity field $v:\mathbb{R}^{n+1}\to \mathbb{R}$. The Lagrangian is a local functional

$$L[q(\cdot,t),v(\cdot,t)]~=~\int d^nx~{\cal L}\left(q(x,t),\partial q(x,t),\partial^2q(x,t), \ldots,\partial^Nq(x,t);\right. $$
$$\left. v(x,t),\partial v(x,t),\partial^2 v(x,t), \ldots,\partial^N v(x,t);x,t\right). $$

The Lagrangian density ${\cal L}$ is a function of these variables. Here $N\in\mathbb{N}$ is some finite order. Moreover, $\partial$ denote partial derivatives wrt. spatial variables $x$, not the temporal variable $t$.

Time $t$ plays the role of a passive spectator parameter, i.e, we may consider a specific Cauchy surface, where time $t$ has some fixed value, and where it makes sense to specify $q(\cdot,t)$ and $v(\cdot,t)$ independently. (If we consider more than one time instant, then the $q$ and $v$ profiles are not independent. See also e.g. this Physics.SE question.)

Weinberg is using the word *functional* because of the spatial dimensions. (In particular, if $n=0$, then Weinberg would have called the Lagrangian $L(q(t),v(t))$ a *function* of the instantaneous position $q(t)$ and the instantaneous velocity $v(t)$.)

It is important to treat $q(\cdot,t)$ (which Weinberg calls $\Psi(\cdot,t)$) and $v(\cdot,t)$ (which Weinberg calls $\dot{\Psi}(\cdot,t)$) for fixed time $t$ as two independent functions in order to make sense of the definition of the conjugate/canonical momentum $p(\cdot,t)$ (which Weinberg calls $\Pi(\cdot,t)$). The definition involves a functional/variational derivative wrt. to the velocity field, cf. eq. (7.2.1) in Ref. 1,

$$\tag{7.2.1} p(x,t)~:=~\frac{\delta L[q(\cdot,t),v(\cdot,t)]}{\delta v(x,t)}.$$

Let us finally integrate over time $t$. The action $S$ (which Weinberg calls $I$) is

$$\tag{7.2.3} S[q]~:=~\int dt~ \left. L[q(\cdot,t),v(\cdot,t)]\right|_{v=\dot{q}}$$

The corresponding Euler-Lagrange equation becomes

$$\tag{7.2.2} \left.\frac{d}{dt} \left(\frac{\delta L[q(\cdot,t),v(\cdot,t)]}{\delta v(x,t)}\right|_{v=\dot{q}}\right)~=~\left. \frac{\delta L[q(\cdot,t),v(\cdot,t)]}{\delta q(x,t)}\right|_{v=\dot{q}}$$

References:

- S. Weinberg,
*The Quantum Theory of Fields,* Vol 1.

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