
Euclidean Spaces - bemeurer
https://meurer.xyz/post/2018-11-18-euclidean-spaces/
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nimish
I'm going to plug Alan Macdonald's Linear and Geometric Algebra book as a
great way to get an intuitive grip on the latter parts of this blog post

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nikofeyn
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.

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llamaz
This 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 that it's more conversational, giving you
the opportunity to add your own insight to the problem.

The problem is that you need to have mulled over the problem for months to
years before you can develop insight.

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canalli
> 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 in handy.
Typically, in linear algebra course one is introduced to norms and their
properties; but analysis doesn't care about this stuff - it's just that LA
ideas are used to further generalize analysis concepts. Fourier analysis (in
mathematically rigorous sense) is introduced relatively late in ones analysis
edjumacation. But the subject is important to engineers and physicists, so
they get to be introduced to Fourier stuff as early as possible, but with much
of the analytic rigor stripped.

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llamaz
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](http://people.math.gatech.edu/~meyer/MA6701/module5.pdf)

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johnklos
This is the first time I've ever seen an actual, legitimate web site using an
.xyz domain.

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tenkabuto
Mastodon community for maths: [https://mathstodon.xyz](https://mathstodon.xyz)

