Infinite-dimensional vector function
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. Example. Set formula_1 for every positive integer formula_2 and every real number formula_3 Then the function formula_4 defined by the formula formula_5 takes values that lie in the infinite-dimensional vector space formula_6 (or formula_7) of real-valued sequences. For example, formula_8 As a number of different topologies can be defined on the space formula_9 to talk about the derivative  of formula_10 it is first necessary to specify a topology on formula_6 or the concept of a limit in formula_12 Moreover, for any set formula_13 there exist infinite-dimensional vector spaces having the (Hamel) dimension of the cardinality of formula_14 (for example, the space of functions formula_15 with finitely-many nonzero elements, where formula_16 is the desired field of scalars). Furthermore, the argument formula_17 could lie in any set instead of the set of real numbers. Integral and derivative. 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.) derivatives (unless, for example, formula_6 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. Derivatives. If formula_19 where formula_6 is a Banach space or another topological vector space then  the derivative of formula_4 can be defined in the usual way: formula_22 Functions with values in a Hilbert space. If formula_4 is a function of real numbers with values in a Hilbert space formula_9 then the derivative of formula_4 at a point formula_17 can be defined as in the finite-dimensional case: formula_27 Most results of the finite-dimensional case also hold in the infinite-dimensional case too, with some modifications. Differentiation can also be defined to functions of several variables (for example, formula_28 or even formula_29 where formula_30 is an infinite-dimensional vector space). If formula_6 is a Hilbert  space then any derivative (and any other limit) can be computed componentwise: if formula_32 (that is, formula_33 where formula_34 is an orthonormal basis of the space formula_6), and formula_36 exists, then formula_37 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 formula_6 too. 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. Crinkled arcs. If formula_39 is an interval contained in the domain of a curve formula_4 that is valued in a topological vector space then the vector formula_41 is called the "chord of formula_4 determined by formula_39". If formula_44 is another interval in its domain then the two chords are said to be "non−overlapping chords" if formula_39 and formula_44 have at most one end−point in common. 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. A "crinkled arc" is an injective continuous curve with the property that any two non−overlapping chords are orthogonal vectors. An example of a crinkled arc in the Hilbert formula_47 space formula_48 is: formula_49 where formula_50 is the indicator function defined by formula_51 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 formula_52 A crinkled arc formula_53 is said to be "normalized" if formula_54 formula_55 and the span of its image formula_56 is a dense subset of formula_12 If formula_58 is an increasing homeomorphism then formula_59 is called a "reparameterization" of the curve formula_60 Two curves formula_4 and formula_62 in an inner product space formula_6 are "unitarily equivalent" if there exists a unitary operator formula_64 (which is an isometric linear bijection) such that formula_65 (or equivalently, formula_66). Measurability. The measurability of formula_4 can  be defined by a number of ways, most important of which are Bochner measurability and weak measurability. Integrals. The most important integrals of formula_4 are called Bochner integral (when formula_6 is a Banach space) and Pettis integral (when formula_6 is a topological vector space). Both these integrals commute with linear functionals. Also formula_71 spaces have been defined for such functions.