what is Norm?
Well, since we’re on the topic of norms, it reminds me of the late Norm Macdonald, one of my favorite comedians. Just like Norm had a unique way of looking at the world, mathematical norms give us a unique perspective on vectors. so anyway, So, what is a mathematical norm?
In mathematics, a norm is a function that assigns a positive value to a vector, representing its magnitude. This concept is closely associated with normed spaces, which are vector spaces equipped with a norm. The term “L” in norms, such as Lp norms, originates from Henri Léon Lebesgue, a mathematician from the early 20th century. He is best known for his PhD thesis on Lebesgue integrals, which, while not widely applicable in everyday scenarios, provided advancements in solving problems related to Riemann integrals, especially when dealing with crazy functions like the Dirichlet function or exploring concepts such as measure theory ,countability and uncountability, which have implications for integrals. This includes intriguing paradoxes like Cantor’s paradox and the Hilbert hotel problem, which challenge our understanding of infinity and set theory.
When we discuss norms, we often refer to normed spaces and unit spheres. A normed space is a vector space equipped with a norm, allowing us to explore the geometry of these spaces. The unit sphere, on the other hand, consists of all vectors with a magnitude of one, serving as a fundamental concept in understanding these spaces.
so first the path is like this: norms -> $L_P$ spaces -> normed vecotr spaces -> hilbert spaces
Sources
Contor’s paradox
- https://www.youtube.com/watch?v=CvalbBGhmW4 (little bit make it more complex than it need
- https://www.youtube.com/watch?v=0HF39OWyl54 (this one mostly mistaken me, actually contor’s argument was make the d numbers digits not be the diangal so it not fit whatever list in whatever order you come up with)
normed vector spaces (didn’t fully read yet)
- https://jonathan-hui.medium.com/vector-space-normed-space-hilbert-space-machine-learning-b43e5d0ac9d3
- https://mbernste.github.io/posts/normed_vector_space/
- https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.020/sauter/ws14/normed-spaces.2014-11-12.pdf
functional analysis (MIT)
- https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/resources/18102-sp21-lecture-1/
wikipedia
- https://en.wikipedia.org/wiki/Lp_space
- https://en.wikipedia.org/wiki/Norm_(mathematics)
a bit about von Neumann
- https://www.sciencedirect.com/science/article/abs/pii/S1355219896000172
- https://medium.com/the-haven/we-are-complaining-that-oppenheimer-did-not-have-john-von-neumann-because-we-are-not-real-acc9581f69b4