5. Normal spaces. Banach spaces

In elementary geometry, the norm of a vector is defined as its length. Vector calculus extends easily to very general functional spaces, and it's useful to be able to use the notion of norm of a function, an operator, a linear form... The structure of normed space that we're now defining is fundamental when we want to apply geometric ideas to analysis.

Definition 1: Let E be a vector space. A normN on E is an application $N : E \to [0, + \infty]$ that verifies the following...

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