LINEAR FUNCTIONALS Meaning and
Definition
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Linear functionals, sometimes also referred to as linear forms or linear maps, are a fundamental concept in linear algebra and functional analysis. A linear functional is a mathematical function or map that takes as input a vector from a vector space and produces a scalar as output.
More precisely, a linear functional f is a mapping from a vector space V to the field of scalars, denoted as F. For every pair of vectors u, v ∈ V and scalar λ ∈ F, the functional satisfies two main properties: linearity and additivity.
Linearity means that the functional preserves the operations of addition and scalar multiplication. In other words, for any vectors u, v ∈ V and scalars λ ∈ F, the functional satisfies the equation f(λu + v) = λf(u) + f(v).
Additivity means that the functional preserves vector addition. It is given by f(u + v) = f(u) + f(v) for all u, v ∈ V.
Linear functionals play a crucial role in mathematical analysis and the study of vector spaces. They are used to define and analyze concepts such as dual spaces, functionals and function spaces, linear transformations, and linear systems. Additionally, they are fundamental tools in optimization theory, mathematical physics, and other branches of mathematics and applied sciences. Linear functionals have a wide range of applications and provide a powerful framework for modeling and analyzing various real-world problems.
Etymology of LINEAR FUNCTIONALS
The word "linear" originated from the Latin word "linearis", which means "pertaining to a line". In mathematics, a linear function is defined as a function that satisfies the properties of additivity and homogeneity. The term "functionals" is derived from the noun "functional", which refers to a mathematical function that takes a function as its argument and returns a scalar value or another function. The combination of "linear" and "functionals" creates the term "linear functionals", which specifically refers to linear mappings from a vector space to its underlying field.
