Mathematical formulation of quantum mechanics

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are those mathematical formalisms that permit a rigorous description of quantum mechanics.

This mathematical formalism uses mainly a part of functional analysis,
especially Hilbert space which is a kind of linear space.

Such are distinguished from mathematical formalisms for physics theories
developed prior to the early 1900s by the use of abstract mathematical structures,
such as infinite-dimensional Hilbert spaces (L2 space mainly), and
operators on these spaces.

In brief, values of physical observables such as energy and momentum were
no longer considered as values of functions on phase space,
but as eigenvalues; more precisely as spectral values of linear operators in Hilbert space.[1]

Wiki

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Formalism is the view that holds that statements of mathematics and logic
can be considered to be statements about the consequences of the manipulation of strings
(alphanumeric sequences of symbols, usually as equations)
using established manipulation rules.