An infinite-dimensional vector function is a function whose values lie in an infinite-dimensional topological vector space, such as a Hilbert space or a Banach space.
Such functions are applied in most sciences including physics.
takes values that lie in the infinite-dimensional vector space
As a number of different topologies can be defined on the space
there exist infinite-dimensional vector spaces having the (Hamel) dimension of the cardinality of
with finitely-many nonzero elements, where
Most theorems on integration and differentiation of scalar functions can be generalized to vector-valued functions, often using essentially the same proofs.
Perhaps the most important exception is that absolutely continuous functions need not equal the integrals of their (a.e.)
is a Hilbert space); see Radon–Nikodym theorem A curve is a continuous map of the unit interval (or more generally, of a non−degenerate closed interval of real numbers) into a topological space.
An arc is a curve that is also a topological embedding.
A curve valued in a Hausdorff space is an arc if and only if it is injective.
is a function of real numbers with values in a Hilbert space
can be defined as in the finite-dimensional case:
Differentiation can also be defined to functions of several variables (for example,
is a Hilbert space then any derivative (and any other limit) can be computed componentwise: if
However, the existence of a componentwise derivative does not guarantee the existence of a derivative, as componentwise convergence in a Hilbert space does not guarantee convergence with respect to the actual topology of the Hilbert space.
Most of the above hold for other topological vector spaces
However, not as many classical results hold in the Banach space setting, for example, an absolutely continuous function with values in a suitable Banach space need not have a derivative anywhere.
Moreover, in most Banach spaces setting there are no orthonormal bases.
is an interval contained in the domain of a curve
[1] Intuitively, two non−overlapping chords of a curve valued in an inner product space are orthogonal vectors if the curve makes a right angle turn somewhere along its path between its starting point and its ending point.
If every pair of non−overlapping chords are orthogonal then such a right turn happens at every point of the curve; such a curve can not be differentiable at any point.
[1] A crinkled arc is an injective continuous curve with the property that any two non−overlapping chords are orthogonal vectors.
A crinkled arc can be found in every infinite−dimensional Hilbert space because any such space contains a closed vector subspace that is isomorphic to
[2] Proposition[2] — Given any two normalized crinkled arcs in a Hilbert space, each is unitarily equivalent to a reparameterization of the other.
are unitarily equivalent if there exists a unitary operator
(which is an isometric linear bijection) such that
can be defined by a number of ways, most important of which are Bochner measurability and weak measurability.
is a Banach space) and Pettis integral (when
Both these integrals commute with linear functionals.