What is Hilbert Space in Mathematics and Quantum Mechanics?

In this episode, we will be covering Hilbert Space and its importance in quantum mechanics. Please cut me some slack because this topic combines abstract algebra with advanced calculus, both of which I do not know. The history of Hilbert Space and the application of it into quantum mechanics should be simple, but its application and derivation surrounding mathematics will likely stump me; I know the English language, but not integrals and multivariable vector calculus. But anyways, let’s explore the foundation of something I do not understand and the foundation of quantum mechanics, Hilbert space. References: Hilbert Space - Quantiki https://www.quantiki.org/wiki/hilbert-spaces Hilbert Space - Wikipedia https://en.wikipedia.org/wiki/Hilbert_space Vector Calculus - Wikipedia https://en.wikipedia.org/wiki/Vector_calculus Euclidean Space - Britannica https://www.britannica.com/science/Euclidean-space Vector Calculus http://hyperphysics.phy-astr.gsu.edu/hbase/vecal.html Quick Hilbert Space Video - Physics World https://www.youtube.com/watch?v=zASzj4DkZXY Hilbert Space - Britannica https://www.britannica.com/science/Hilbert-space Functional Analysis - Britannica https://www.britannica.com/science/functional-analysis-mathematics Vector Fields - Wikipedia https://en.wikipedia.org/wiki/Vector_field Probability Density Function - Wikipedia https://en.wikipedia.org/wiki/Probability_density_function