Why Functional is Named Functional
The term “functional analysis” is indeed intriguing, especially when considering the word “functional” instead of merely “function analysis.” At first glance, “functional” appears to be an adjective form of “function,” suggesting meanings like “function-like” or “pertaining to functions.” This notion can also be found in another name for functionals, “generalized functions.” The question arises as to why these are not simply called functions. To understand this, let’s look at the general definition of a functional:
For a vector space $X$, a function $f$ called a functional is defined as follows:
$$ f : X \to \mathbb{C} $$
This definition might prompt the question, “If $f$ is a function according to the above definition, why is it named a functional?” Although it’s understandable to assign a special name to functions satisfying certain conditions, the reason behind the specific choice of “functional” (implying something function-like) may not be immediately clear.
To grasp why something would be labeled as “function-like” rather than a function, it’s essential to consider the context in which functional analysis emerged. People learning mathematics today understand functions as follows:
A correspondence $f$ is said to exist from $X$ to $Y$ if, for every $x_{1}, x_{2} \in X$, there exist $f(x_{1})$ and $f(x_{2})$ that satisfy $x_{1} = x_{2} \implies f(x_{1}) = f(x_{2})$.
$$ f : X \to Y $$
When rigorously defined using set theory, it becomes:
Given two non-empty sets $X$ and $Y$, a binary relation $f \subset (X,Y)$ is called a function if it satisfies the following, represented as $f : X \to Y$:
$$ (x ,y_{1}) \in f \land (x,y_{2}) \in f \implies y_{1} = y_{2} $$
As seen in these definitions, there are no specific conditions on sets $X$ and $Y$; whether they are sets of numbers or function spaces does not matter. However, to mathematicians in the late 19th century, a function was not perceived this way. They thought of functions primarily as mappings from values to values, essentially as formulas that provide one value from another, similar to how functions are introduced in middle school.
This view was somewhat natural because the rigorous definition of functions, as shown above, was developed through set theory, which was founded by Cantor, born in 1845. Thus, it’s not surprising that mathematicians up to the early 20th century considered functions mainly in terms of numerical formulas. The term “function” itself suggests a certain functionality or operation.
Consider the following function:
For a differentiable function $f$ over a closed interval $[a,b]$, the length of the curve $y=f(x)$ is defined as follows:
$$ L(f)=\int_{a}^{b} \sqrt{1+ f^{\prime}(x)^{2}}dx $$
To the mathematicians of the time, $L$ was not considered a function because it mapped functions to values, not values to values. Hence, $L$ could be termed a “function of functions,” but since it was not strictly a function, there was ambiguity in terminology. Volterra referred to them as “functions of lines,” and the French mathematician Hadamard proposed calling these “function-like functions of functions” fonctionnelles. This term later became “functional” in English.
After functions were rigorously defined using set theory, functionals became considered functions too. However, the term “functional” continues to be used, particularly because it clearly indicates that the domain is a space of functions, much like how a “collection” or “family” refers to a set of sets. Despite the conceptual overlap between functionals and functions, the term “functional” has persisted, likely to avoid confusion. The field of study being named functional analysis likely also played a role. Over time, the term “functional” has evolved to refer to mappings from vector spaces to complex number spaces.
Distribution Theory
As discussed, the term “functional” was originally coined to describe entities that were like functions but not exactly functions. However, once functions were defined through set theory, functionals became recognized as functions. Interestingly, functionals ended up being used to describe entities that truly were “not functions but function-like.” The Dirac delta function, first conceptualized by Poisson and Cauchy during their study of Fourier analysis and later popularized by the theoretical physicist Paul Dirac in quantum mechanics, is an example of such an entity. Its naive definition satisfies certain conditions that imply divergence, meaning the delta function is not strictly a function but rather a state or condition.
In 1950, after 15 years of research, the French mathematician Laurent-Moise Schwartz rigorously defined the delta function in the book “Theorie des distributions.” He introduced the concept of smooth functions called test functions and their space, denoted as $\mathcal{D}$. Distributions are mappings from $\mathcal{D}$ to $\mathbb{C}$ and are considered functionals. Although initially functional referred to entities not traditionally regarded as functions, it eventually came to be used in developing a theory for entities that are not functions in the conventional sense but are treated as such, marking a fascinating turn of events.