On normed spaces and analysis in normed spaces

In this thesis, normed spaces and analysis in normed spaces have been researched and studied. Normed spaces have been defined and examples of normed spaces are indicated. The features of analysis in normed spaces are presented. The convergence of sequences in normed spaces is defined. The properties of covergent sequences are demonstrated. The varieties of the convergent sequences in seperate normed spaces are given. The analysis problems in normed spaces are handled. The definitions of open and closed sets in normed spaces are given and examples are given. The equivalence of the norms infinite dimensional spacesisproved. Riesz's Lemma is written and proved. The approximation problem with the elements of subspaceisstudied. Theorems about the best approximation are written and proved. The definition of dense linear manifold in everywhere in normed spacesisgiven and exemplified. Definitions of average and interrupt functions are given and the practices of them are indicated.