Don't let the terminology intimidate you. The interesting ideas in quantum computing are far more dependent upon a foundation in linear algebra rather than a foundation in mathematical analysis.
When I started out, I was under the assumption that I had to understand at least the undergraduate real analysis curriculum before I could grasp quantum algorithms. In reality, for the main QC algorithms you see discussed, you don't need to understand completeness; you can just treat a Hilbert space as a finite-dimensional vector space with a complex inner product.
For those unfamiliar with said concepts from linear algebra, there is a playlist [1] often recommended here which discusses them thoroughly.
Yeah all the names and terminology really do make it seem harder than it is. Took me a long time and I’m still learning. 2d Hilbert space is same as 2d Euclidean space but each dimension has 2 degrees of freedom (real + imaginary). Might even think of it as 4d space, for vector imagining purposes, but that would probably be wrong and someone would call you out
Better start with Simon's algorithm (solving Simon's problem) [0]; it already contains a lot of ideas that you need to understand Shor's algorithm, while not having a lot of technicalities. Then progress to Shor's algorithm, and then to Kitaev's algorithm [1] (link from [2]). The latter solves the Abelian stabilizer problem - this problem contains the more abstract mathematical essence of a lot of quantum algorithms.
When I started out, I was under the assumption that I had to understand at least the undergraduate real analysis curriculum before I could grasp quantum algorithms. In reality, for the main QC algorithms you see discussed, you don't need to understand completeness; you can just treat a Hilbert space as a finite-dimensional vector space with a complex inner product.
For those unfamiliar with said concepts from linear algebra, there is a playlist [1] often recommended here which discusses them thoroughly.
[1] https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...