The notion of linear space came into existence in the effort to obtain a unifying principle for certain algebraic phenomena. After realizing that constructions of a “linear nature” repeat themselves in several situations, it became appropriate to adopt an axiomatic approach. It is the axiomatic approach that gives rise to the notion of linear space. The advantage of this modern and largely used attitude is that one obtains results simultaneously applicable in apparently totally different areas. The effort needed for proving these more general results is, in fact, the same as the effort needed for proving them in a particular model. The alleged flaw seems to be a higher degree of abstraction. A beginner may indeed experience a certain inconvenience, particularly in the case when the secondary school in which he studied neglected the preparation of “mathematical thinking”. But this can be overcome. Sooner or later the hardworking student will appreciate the axiomatic approach and begin to like it. There is no doubt that he (she) will profit from his (her) skill acquired in the axiomatic approach when confronted with the algebraic problems of engineering sciences. We will define the notion of linear space first and then we will illustrate this notion by several examples. Before, let us make the following conventions. The letter R will denote the set of all real numbers understood with their usual algebraic structure. The letter N will stand for the set of all natural numbers. The symbols ∪, ∩, − will denote the standard set theory operations of union, intersection, and complementation. The symbol P × Q will denote the cartesian product of the sets P and Q. Finally, the logical quantifiers ∀ (“for every”) and ∃ (“there exists”), as well as the logical implication ⇒, will be occasionally used to simplify the formulation of the results. The ends of proofs will be denoted by ✷ .
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