Linear Algebra and Analysis approach mathematical abstraction from two seemingly different perspectives. However, once we start talking about normed linear spaces or, more concretely, Hilbert spaces, these two subjects readily connected to each other and the new theory ultimately became the playground for physicists in the 1900s. In this talk, we present some background material on both linear algebra and elementary analysis, discuss their roles within the concept of a Hilbert Space, and why a Hilbert space is important to Quantum Mechanics.