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Liu, H. Infinite–dimensional Vector Function. Encyclopedia. Available online: https://encyclopedia.pub/entry/34654 (accessed on 02 December 2023).
Liu H. Infinite–dimensional Vector Function. Encyclopedia. Available at: https://encyclopedia.pub/entry/34654. Accessed December 02, 2023.
Liu, Handwiki. "Infinite–dimensional Vector Function" Encyclopedia, https://encyclopedia.pub/entry/34654 (accessed December 02, 2023).
Liu, H.(2022, November 15). Infinite–dimensional Vector Function. In Encyclopedia. https://encyclopedia.pub/entry/34654
Liu, Handwiki. "Infinite–dimensional Vector Function." Encyclopedia. Web. 15 November, 2022.
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.

infinite–dimensional infinite-dimensional

## 1. Example

Set $\displaystyle{ f_k(t) = t/k^2 }$ for every positive integer $\displaystyle{ k }$ and every real number $\displaystyle{ t. }$ Then the function $\displaystyle{ f }$ defined by the formula $\displaystyle{ f(t) = (f_1(t), f_2(t), f_3(t), \ldots)\, , }$ takes values that lie in the infinite-dimensional vector space $\displaystyle{ X }$ (or $\displaystyle{ \R^{\N} }$) of real-valued sequences. For example, $\displaystyle{ f(2) = \left(2, \frac{2}{4}, \frac{2}{9}, \frac{2}{16}, \frac{2}{25}, \ldots\right). }$

As a number of different topologies can be defined on the space $\displaystyle{ X, }$ to talk about the derivative of $\displaystyle{ f, }$ it is first necessary to specify a topology on $\displaystyle{ X }$ or the concept of a limit in $\displaystyle{ X. }$

Moreover, for any set $\displaystyle{ A, }$ there exist infinite-dimensional vector spaces having the (Hamel) dimension of the cardinality of $\displaystyle{ A }$ (for example, the space of functions $\displaystyle{ A \to K }$ with finitely-many nonzero elements, where $\displaystyle{ K }$ is the desired field of scalars). Furthermore, the argument $\displaystyle{ t }$ could lie in any set instead of the set of real numbers.

## 2. 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, $\displaystyle{ X }$ 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.

### 2.1. Derivatives

If $\displaystyle{ f : [0,1] \to X, }$ where $\displaystyle{ X }$ is a Banach space or another topological vector space then the derivative of $\displaystyle{ f }$ can be defined in the usual way: $\displaystyle{ f'(t) = \lim_{h\to 0}\frac{f(t+h)-f(t)}{h}. }$

#### Functions with values in a Hilbert space

If $\displaystyle{ f }$ is a function of real numbers with values in a Hilbert space $\displaystyle{ X, }$ then the derivative of $\displaystyle{ f }$ at a point $\displaystyle{ t }$ can be defined as in the finite-dimensional case: $\displaystyle{ f'(t)=\lim_{h\to 0} \frac{f(t+h)-f(t)}{h}. }$ 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, $\displaystyle{ t \in R^n }$ or even $\displaystyle{ t\in Y, }$ where $\displaystyle{ Y }$ is an infinite-dimensional vector space).

If $\displaystyle{ X }$ is a Hilbert space then any derivative (and any other limit) can be computed componentwise: if $\displaystyle{ f = (f_1,f_2,f_3,\ldots) }$ (that is, $\displaystyle{ f = f_1 e_1+f_2 e_2+f_3 e_3+\cdots, }$ where $\displaystyle{ e_1,e_2,e_3,\ldots }$ is an orthonormal basis of the space $\displaystyle{ X }$), and $\displaystyle{ f'(t) }$ exists, then $\displaystyle{ f'(t) = (f_1'(t),f_2'(t),f_3'(t),\ldots). }$ 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 $\displaystyle{ X }$ 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.

### 2.2. Crinkled Arcs

If $\displaystyle{ [a, b] }$ is an interval contained in the domain of a curve $\displaystyle{ f }$ that is valued in a topological vector space then the vector $\displaystyle{ f(b) - f(a) }$ is called the chord of $\displaystyle{ f }$ determined by $\displaystyle{ [a, b] }$. If $\displaystyle{ [c, d] }$ is another interval in its domain then the two chords are said to be non−overlapping chords if $\displaystyle{ [a, b] }$ and $\displaystyle{ [c, d] }$ 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 $\displaystyle{ L^2 }$ space $\displaystyle{ L^2(0, 1) }$ is: \displaystyle{ \begin{alignat}{4} f :\;&& [0, 1] &&\;\to \;& L^2(0, 1) \\[0.3ex] && t &&\;\mapsto\;& \mathbb{1}_{[0,t]} \\ \end{alignat} } where $\displaystyle{ \mathbb{1}_{[0,\,t]} : (0, 1) \to \{0, 1\} }$ is the indicator function defined by $\displaystyle{ x \;\mapsto\; \begin{cases}1 & \text{ if } x \in [0, t]\\ 0 & \text{ otherwise } \end{cases} }$ 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 $\displaystyle{ L^2(0, 1). }$ A crinkled arc $\displaystyle{ f : [0, 1] \to X }$ is said to be normalized if $\displaystyle{ f(0) = 0, }$ $\displaystyle{ \|f(1)\| = 1, }$ and the span of its image $\displaystyle{ f([0, 1]) }$ is a dense subset of $\displaystyle{ X. }$

Proposition — Given any two normalized crinkled arcs in a Hilbert space, each is unitarily equivalent to a reparameterization of the other.

If $\displaystyle{ h : [0, 1] \to [0, 1] }$ is an increasing homeomorphism then $\displaystyle{ f \circ h }$ is called a reparameterization of the curve $\displaystyle{ f : [0, 1] \to X. }$ Two curves $\displaystyle{ f }$ and $\displaystyle{ g }$ in an inner product space $\displaystyle{ X }$ are unitarily equivalent if there exists a unitary operator $\displaystyle{ L : X \to X }$ (which is an isometric linear bijection) such that $\displaystyle{ g = L \circ f }$ (or equivalently, $\displaystyle{ f = L^{-1} \circ g }$).

### 2.3. Measurability

The measurability of $\displaystyle{ f }$ can be defined by a number of ways, most important of which are Bochner measurability and weak measurability.

### 2.4. Integrals

The most important integrals of $\displaystyle{ f }$ are called Bochner integral (when $\displaystyle{ X }$ is a Banach space) and Pettis integral (when $\displaystyle{ X }$ is a topological vector space). Both these integrals commute with linear functionals. Also $\displaystyle{ L^p }$ spaces have been defined for such functions.

### References

1. Halmos 1982, pp. 5−7.
2. Halmos 1982, pp. 5−7,168−170.
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