Help with understanding $L^2(\mathbb{R},\mathbb{R}^n)$ manifolds (?)

Question

I've recently started studying the square root velocity function representation of real functions and curves. Authors of the papers I have do not explain the underlying mathematical concepts and I couldn't google any relevant texts. For example they consider a certain subset of $L^2(\mathbb{R},\mathbb{R}^n)$ (the vector-valued analogue of the standard $L^2(\mathbb{R})$), namely a unit sphere under a certain norm. Now they take a manifold-like structure of that subspace for granted and start constructing tangent spaces, providing Riemannian structure and talking about geodesics.

I don't understand how the manifold structure here is obvious as the commonly known differential geometry theory applies only to finite-dimensional spaces. This one is clearly infinitely-dimensional. Could you suggest me any introductory reading that could help me understand it?

EDIT: After seeing that there is a very similar questions with answers I have one thing to add: the accepted answer gives a link to an interesting book but it's unclear to me if $L^2(\mathbb{R},\mathbb{R}^n)$ is convenient.

Answer 1

Some suggestions you may want to check (assuming there is a math library available to you):

Abraham, Marsden, Ratiu: Manifolds, Tensor Analysis and Applications.

Deimlings Nonlinear Functional Analyis.

Serge Lang: Real Analysis

Klingenbergs Differential Geometry. The deGruyter book.

Each book covers different aspects. There may be more recent introductory texts, but regardless of that: judging from what you explain you already know: a lot of time and patience.