this is just a collection of basic facts and definitions with bad language. for example, inner product space is the correct nomenclature. euclidean space refers to an inner product space with the euclidean norm, i.e., the dot product. euclidean space should not be used to refer to a general inner product space.
Euclidean Spaces
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Re: Euclidean Spaces
#12This particular topic is something that all 19 year old electrical/mechatronic engineering students at my university in Australia learn, so it's probably a standard topic around the world (I think it's used to understand Fourier analysis in more advanced courses). Currently the post reads similar to what most readers would have encountered over the course of a 2 hour lecture, so my advice would be to vary the tone so…
> so it's probably a standard topic around the world (I think it's used to understand Fourier analysis in more advanced courses). These notes are contained in a chapter or two of any standard linear algebra textbook. This can serve as a background when studying analysis. Analysis starts with considering the real line first, then moves on to the metric spaces, then the normed spaces etc. That's when this stuff comes i…
To add to what you said:
In Australia the follow on course after second year analysis is a course on topology, metric spaces, and basic functional analysis. Here you learn about norms from an analysis point of view and its relationship to topology (e.g. the euclidean norm induces the euclidean topology, which is a set of of open balls satisfying some properties).
One specific topic in the blog post, "best approximation" [1], is used to add some amount of rigour to the engineer's version of fourier analysis.
[1]: this is the first ref I could find on google: http://people.math.gatech.edu/~meyer/MA6701/module5.pdf