Short answers, pulled from the story.
A vector space is a set whose elements, called vectors, can be added together and multiplied by numbers called scalars, subject to eight axioms covering associativity, commutativity, identity elements, inverses, and distributivity. Real vector spaces use real-number scalars; complex vector spaces use complex-number scalars. More generally, scalars can be elements of any field.
Giuseppe Peano gave the first modern definition of vector spaces and linear maps in 1888, calling them linear systems. His axiomatization allowed for infinite-dimensional vector spaces. In 1897, Salvatore Pincherle adopted Peano's axioms and made initial inroads into the theory of infinite-dimensional vector spaces.
The dimension of a vector space is the cardinality shared by all of its bases. A basis is a set of linearly independent vectors that spans the entire space. All bases of a given vector space have the same cardinality, a fact guaranteed by the dimension theorem for vector spaces.
Banach spaces, introduced by Stefan Banach, are complete normed vector spaces, meaning every Cauchy sequence converges to a limit within the space. Hilbert spaces, named for David Hilbert, are complete inner product spaces. Both were formalized around 1920 and are central objects in functional analysis.
The solutions to the time-dependent Schrodinger equation in quantum mechanics are called wavefunctions and live in a Hilbert space. Definite values of physical properties such as energy and momentum correspond to eigenvalues of linear differential operators, and the associated wavefunctions are the eigenstates of those operators.
Around 1636, Rene Descartes and Pierre de Fermat founded analytic geometry by identifying solutions to equations of two variables with points on plane curves. This coordinate approach was an early step toward the concept of vector spaces, which generalize Euclidean vectors and the geometry of the plane and three-dimensional space.