Incoming links: Norm (mathematics)
Metric space vs normed vector space vs inner product space Updated 2024-12-15 +Created 1970-01-01
TODO examples:
- metric space that is not a normed vector space
- norm vs metric: a norm gives size of one element. A metric is the distance between two elements. Given a norm in a space with subtraction, we can obtain a distance function: the metric induced by a norm.
Some sources say that this is just the part that says that the norm of a function is the same as the norm of its Fourier transform.
Others say that this theorem actually says that the Fourier transform is bijective.
The comment at math.stackexchange.com/questions/446870/bijectiveness-injectiveness-and-surjectiveness-of-fourier-transformation-define/1235725#1235725 may be of interest, it says that the bijection statement is an easy consequence from the norm one, thus the confusion.
TODO does it require it to be in as well? Wikipedia en.wikipedia.org/w/index.php?title=Plancherel_theorem&oldid=987110841 says yes, but courses.maths.ox.ac.uk/node/view_material/53981 does not mention it.
The orthogonal group is the group of all matrices that preserve the dot product Updated 2024-12-15 +Created 1970-01-01
When viewed as matrices, it is the group of all matrices that preserve the dot product, i.e.:This implies that it also preserves important geometric notions such as norm (intuitively: distance between two points) and angles.
This is perhaps the best "default definition".