funktionaalit

Funktionaalit are mathematical mappings from a vector space to the field of scalars over which the space is defined. A funktionaali f: V → F assigns to each vector a scalar. If f satisfies linearity, f(ax + by) = a f(x) + b f(y) for all vectors x, y in V and scalars a, b, it is called a lineaarinen funktionaali (linear functional). The set of all continuous linear functionals on a normed space X is called the dual space, denoted X*. In finite-dimensional spaces, every linear functional is continuous; in infinite-dimensional spaces, continuity is an additional condition with analytic significance.

Examples include functionals defined by inner products: in a Hilbert space H, for a fixed y, the map f_y(x) = ⟨x, y⟩ is a continuous linear functional. In finite-dimensional Euclidean spaces, the standard coordinate functionals f_i(x) = x_i are linear functionals. The dual space concept generalizes to various settings, with different continuity and topological requirements depending on the space.

Key results include the Riesz representation theorem, which states that every continuous linear functional on a Hilbert space H can be uniquely represented as f(x) = ⟨x, y⟩ for some y in H. This bridges functionals with elements of the space itself and underpins many duality arguments in analysis.

Applications of funktionaalit appear in duality theory, optimization, and analysis, including L^p spaces and Fourier analysis, where dual functionals provide important tools for characterizing limits, integrals, and transforms. While often studied in the linear and continuous setting, the broader notion of a functional can also refer to non-linear maps in other contexts.