Linear functional

In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. Specifically, if V is a vector space over a field k, then a linear functional is a linear function from V to k.

Related Topics:
Linear algebra - Mathematics - Linear function - Scalar - Vector space - Field

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The set of all linear functionals from V to k, HomK(V,k), is itself a k-vector space. This space is called the dual space of V. If V is a topological vector space the space of continuous linear functionals, the continuous dual, is often simply called the dual space. If V is a Banach space then so is its continuous dual.

Related Topics:
Dual space - Topological vector space - Banach space

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Linear functionals first appeared in functional analysis, the study of vector spaces of functions. A typical example of a linear functional is integration: the linear transformation

Related Topics:
Functional analysis - Integration

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:f mapsto int_a^b f(x), dx

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is a linear functional from the space of integrable functions to the reals.

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Linear functionals are particularly important in quantum mechanics. Quantum mechanical systems are represented by Hilbert spaces, which are isomorphic to their own dual spaces. A state of a quantum mechanical system can be identified with a linear functional. For more information see bra-ket notation.

Related Topics:
Quantum mechanics - Hilbert space - Bra-ket notation

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