holysword

If anything, Hilbert spaces are "well-behaved" - that's why I asked targeting those spaces instead of something more complicated. But if you want to simplify, we could go with finite-dimensional Hilbert spaces. As I mentioned, "I don't even know how to ask this", I understand the concept of limits, but I've learned it in another language. I mentioned "I'm not a maths student" to clarify that I'm not deeply interested in the intricacies of the very concept of limit, but more into a more pragmatic approach - as an engineer, one might say. Notice I asked for references, not to solve a particular problem. This page: http://tutorial.math.lamar.edu/Classes/CalcI/LimitProofs.aspx contains a series of limit identities, but nothing like I showed. Another example that consider is [latex]\lim_{x\rightarrow a} f(x) \stackrel{?}{=} \lim_{x\rightarrow \frac{a}{b}} f(bx)[/latex] is that always true, regardless of [latex]f(x)[/latex]?

I don't think that examples are a safe way to prove things. No, I cannot. I'm not a maths student. Can you enlighten me please?

I'm asking it here because I don't even know HOW to ask this! Basically, I'm looking for some identities regarding the "tends to" part of the limit. So for instance: [latex]\lim_{x\rightarrow b} f(x) = \lim_{x\rightarrow 0} f(x+b)[/latex] Is the above expression correct? If so, is it always correct, at least for Hilbert spaces? What else can you do with the "tends to" part, can you rename variables there? References are welcome.