2. Hilbert spaces

2.1 The basics

Natural functional spaces are of infinite dimension. It is therefore essential to consider infinite-dimensional spaces when applying ideas from geometry or linear algebra to analysis. Among these, Hilbert spaces, which we will now define, occupy a central position.

Let E be a real vector space. A scalar product on E is an application $〈\mathrm{.,.}〉$ of E × E in $ℝ$ which verifies for all vectors x, y, z and any scalar λ :

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