![]() One addition for the scalar field and another for the set of vectors. These two mathematical structures are entangled with each other via (ii) and (iii). The elements of are called scalars and the elements of are called vectors.Įvery vector space consists of a field and an Abelian group. (ii) Compatibility of scalar multiplication with field multiplication: Such that the following axioms are fulfilled. A vector space is a set of vectors along with an operationĪnd a function, called scalar multiplication, Let us start with the two pivotal definitions of this post. For those who have no exposure to functional analysis, the introduction series to functional analysis provided by The Bright Side of Mathematics on YouTube or one of the book from the literature might help to get you up to speed. ![]() The branch of math that studies function spaces is called functional analysis. ![]() We will, however, focus on mechanics of a function space without diving too deep into the realm of infinite dimensional vector spaces and its specifics. This raises several challenges since general function spaces are infinite dimensional and concepts like basis and linear independence might be reconsidered. In this post, we study specific vector spaces where the vectors are not tuples but functions. This algebraic structure has appeared in many real world problems and is therefore known for centuries. ![]() Vector spaces are one of the most fundamental and important algebraic structures that are used far beyond math and physics. ![]()
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