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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).
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 ...
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.
Any (algebraic) linear functional on a vector subspace can be extended to the whole space; for example, the evaluation functionals described above can be extended to the vector space of polynomials on all of . However, this extension cannot always be done while keeping the linear functional continuous.
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. ...
Every norm, seminorm, and real linear functional is a sublinear function.The identity function on := is an example of a sublinear function (in fact, it is even a linear functional) that is neither positive nor a seminorm; the same is true of this map's negation . [5] More generally, for any real , the map ,: {is a sublinear function on := and moreover, every sublinear function : is of this ...
1. The continuous dual of a topological vector space is the vector space of all the continuous linear functionals on the space. 2. The algebraic dual of a topological vector space is the dual vector space of the underlying vector space.
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.