I read a sentence from a text. I somehow can understand the sentence, but I found that I couldn't analyse the structure of it:-(
Included among such Lipschitz paths are all piecewise smooth paths, for which the generalized definitions of path integrals (with respect to Lipschitz paths) are thus seen to reduce to the definitions given earlier (with respect to piecewise smooth paths)
In this sentence, the words in parenthesis are added by myself, i.e., the original sentence doesn't contains those words. I added them to provide context for you. 'Lipschitz path' and 'piecewise smooth path' are two concepts in math, on both of which we can define 'path integrals'. I think the real meanings of those math concepts don't matter. You can see it this way that 'Lipschitz paths' is a larger class of 'paths' which contains 'piecewise smooth paths'. I also think I can understand the sentence right. It means,
Lipschitz paths included all piecewise smooth paths (in other words, piecewise smooth paths are a special class of Lipschitz paths), so the definition of path integrals on Lipschitz paths can be seen to reduce to the definition of path integrals on piecewise smooth paths given earlier.
After transformation, I found that I couldn't find a subject or object that 'for which' refers to in the latter sentence. But 'for which' must refer to something so that we can transform the second part of the latter sentence to a clause in the first sentence, right?
In my opinion, 'for which' is a bit redundant, or it just means 'because of which'?
Best Answer
Sina is correct that "for which" refers to "all piecewise smooth paths", answering your first question. Specifically, "which" refers to the piecewise smooth paths.
For your second question, "for which" does not mean "because of which". The latter part of the sentence is a description (the generalized definitions... are thus seen to...), and "for which" connects that description to the noun that is described (all piecewise smooth paths, as above). I wouldn't say that "for which" is redundant here because without it the rest sentence as written would be difficult to interpret. But the sentence can be re-written, as in your transformation, in ways that do not require the "for which" phrase at all to connect sentence elements together.