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This is an example of a non-linear functional. The Riemann integral is a linear functional on the vector space of functions defined on [a, b] that are Riemann-integrable from a to b. In mathematics, a functional is a certain type of function. The exact definition of the term varies depending on the subfield (and sometimes even the author).
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, or topology) and the linear functions defined on these spaces and suitably respecting these structures.
In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral theory.
Functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions. Functional integrals arise in probability, in the study of partial differential equations, and in the path integral approach to the quantum mechanics of particles and fields.
A typical example of a linear functional is integration: the linear transformation defined by the Riemann integral = is a linear functional from the vector space [,] of continuous functions on the interval [,] to the real numbers.
Since positive linear functionals are bounded, the equivalence classes of the net {} converges to some vector in , which is a cyclic vector for . It is clear from the definition of the inner product on the GNS Hilbert space H {\displaystyle H} that the state ρ {\displaystyle \rho } can be recovered as a vector state on H {\displaystyle H} .
A formula to determine functional derivatives for a common class of functionals can be written as the integral of a function and its derivatives. This is a generalization of the Euler–Lagrange equation : indeed, the functional derivative was introduced in physics within the derivation of the Lagrange equation of the second kind from the ...
Consider, as an example of , the C*-algebra of complex square matrices with the positive elements being the positive-definite matrices. The trace function defined on this C*-algebra is a positive functional, as the eigenvalues of any positive-definite matrix are positive, and so its trace is positive.
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