I am interested in whether or not the typical vector bundle construction can be extended to cases where each fiber is an infinite dimensional vector space, possibly with other structure associated with it. I have tried to search along these lines but I am not sure if the answers I have found fully answer my question.
This is motivated by the following problem, for context. I have a smooth manifold $M$ (actually, in my case $M$ is just $\mathbb{R}^n$ so it is trivially a manifold). At each point in $M$, I want to be able to associate a function $f: [a,b] \to \mathbb{R}$ such that I have a "function field" on the manifold. If I were dealing with finite dimensional vectors as opposed to functions, I could do this by defining, for example, the tangent bundle $TM$ and vector fields would be a smooth section of the bundle. To provide more context, I have also have a functional $L(v)[\cdot]$ indexed by $v \in \mathbb{R}^n$, and I want to choose the specific $f$ at each point that will maximize this functional. This seems to me to be defining a section of a "function bundle" of sorts, which would be useful to me as I am hoping to be able to compute information about the $f$ at nearby points given the $f$ associated with one point, i.e. to "move along the section" in a well defined way.
I apologize if this is too vague of a question. I have spent some time searching and have not encounter a construct quite like this. I am aware that you can have Banach bundles in which each fiber is infinite dimensional, but the examples of that I have seen are such that both the base space and the fibers are infinite dimensional Banach spaces. In my case the base space is finite dimensional. I would appreciate even just a pointer in the right direction, i.e. some sources or keywords to follow up on. Thank you very much.
Generally speaking, bundles of any sort of object are a way of "smoothly parameterizing" a family of such objects. In differential geometry, the topology and geometry of the collection of all such parametrized families is interesting to study. What you are asking about could be fairly regarded as simply a family of functions that is smoothly parametrized by a parameter $t\in\mathbb R^d$. Sometimes the dependence on the parameter is tucked away behind a semicolon, or used as a subscript, as in $$ f(x;t)\quad\text{or}\quad f_t(x) \qquad (t\in\mathbb R^d). $$ We see this kind of thing fairly commonly on this site when someone asks about Feynman's trick to compute an integral $I=\int f(x)\,dx$ by introducing a parameter $t$ and a parametrized family $\{f_t\}$ such that we can realize $$ I = \frac{d}{dt}\bigg|_{t=t_0}I(t) := \frac{d}{dt}\bigg|_{t=t_0}\int f_t(x)\,dx. $$ You could fairly regard the family $\{f_t\}$ as a "section" of a bundle over the parameter domain $t$, but whether that is useful or not is probably context-dependent.
Another canonical way that these come up is as solutions of partial differential equations. Say we have some PDE $\partial_t u = Lu$ for some operator $L$. Then a solution $u(t) := e^{tL}u_0$ can in some sense be regarded as a path in some infinite-dimensional vector space $V$ of functions (or equivalently as a section of a trivial bundle $I_t\times V\to I_t$, if you like). The study of the mapping properties of "the solution map" $S(t)\colon u_0\mapsto e^{tL}u_0$ is important and natural in PDE.