Teorema espectral para operadores simétricos

Abstract

In the present work we approach one of the most important theorems of linear algebra, called spectral theorem. With the aim of demonstrating it for symmetric operators in IR^n. In didactic terms, we started the work by bringing basic notions, such as: linearly dependent vectors, inner product in the vector space in which we emphasize the properties of the inner product, the Cauchy-Schwarz inequality in the vector space, the superiorly and inferiorly bounded sets and the Axiom Completeness or Dedekind's Postulate. Later, we deal with sequences and discuss limits of sequences and subsequences. Then, we comment on the monotone sequences that can be of the type: increasing, decreasing, non-increasing and non-decreasing. Finally, we deal with the enclosing intervals with the idea of ​​presenting one of the most important theorems for the demonstration of the spectral theorem, which is the Bolzano-Weierstrass theorem in the real case. Throughout the work we present examples for a better understanding of the concepts covered.