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This is very interesting. Thank you. What book would you recommend for this discussion?
product space. Two examples are the real line and the 3 space we live in (at least locally ignoring general relativity). But the biggie points about the Hilbert space definition are (A) the other, somewhat astounding (vector space), examples and (B) how much can do in Hilbert space that is close to good old high school plane (2D) and solid (3D) geometry and close to good, old freshman calculus with limits, convergenc…
I have a utilitarian understanding of mathematics (vector spaces, SVD, orthogonality, invariances, etc.) and over time appreciating the underlying characteristics/relationships, which I recently got a taste of from T. Wickens, The Geometry of Multivariate Statistics.
I look forward to understanding measure theory and related math.