I am trying to see why the $\{x\in \mathbb{R}^{d}:|x^{2}|\leq \varepsilon\}$ (equipped with a ring, say sheaf of continuous functions on that set) is a scheme i.e.to which affine schemes is it locally isomorphic to; to which spectrum of a commutative ring?
My background is in analysis, so I would appreciate any comments/remarks from there.
I am trying to find a simple example of a scheme with an analytic flavour to it.
Attempt
I am thinking of schemes as manifolds with the parameter space being algebraic varieties of polynomials i.e. locally viewed as the locus of a system of polynomials.
The "thickened variety" is $\{x\in \mathbb{R}^{d}:|x^{2}|\leq \varepsilon\}=B_{0}(\sqrt{\varepsilon})$. So is one possible "gluing" the locus of the equations $\{x_{1}^{2}+...+x_{d}^{2}=r^{2}\}$ for all $r\leq \sqrt{\varepsilon}$?
I think the issue your idea of a scheme. As you say, each scheme is a locally ringed space (topological space+sheaf of functions), but everything must be defined algebraically.
The functions on a scheme are only the algebraic ones (e.g. polynomials), which is a much smaller ring than all continuous functions. Also, the topological space must be cut out by polynomials using equalities only, we cannot use inequalities. Because you have less choice, everything is more rigid in algebraic geometry.
If we blindly take the spectrum of something like $C[0,1]$, we get something pathological that doesn't reflect $[0,1]$.
I'm no expert on this stuff, but I think the "Tate algebra" and "rigid analytic varities" are closer to what you want. Here, we can define a disk of radius $r$, but it's over a nonarchimedian field instead of $\mathbb{R}$.