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A hadronic form factor parametrizes a current matrix element between hadron states, separating allowed Lorentz covariant tensor structures from scalar-valued functions of Lorentz invariant combinations of momentum transfer. These functions encode composite structure and strong interaction dynamics. Equal kinematic phase space does not imply equal form factors for different weak interaction transitions.

A real-valued function takes each function input to a real number. Its codomain can be all of $\mathbb{R}$ or a specified subset. Thus $f(x)=x^2$ on the real line is a real-valued function and has image $[0,\infty )$, whether the declared target is that interval or all of $\mathbb{R}$.

The inputs need not be real numbers: the cardinality of a finite set and the squared norm of a vector are also real-valued functions. The adjective describes outputs, while scalar-valued function allows outputs in other fields, notably the complex numbers.

A scalar-valued function is a function whose codomain is a field, typically the real numbers or complex numbers. Its output is a single scalar, while its input can belong to an arbitrary set. For example, the square of the norm of a vector is real-valued. In a relativistic application, being scalar-valued alone does not establish Lorentz invariance: the transformation of the input and of the complete quantity must also be specified.

For a scalar-valued function $f$ on a topological space $X$, its support isThe closure matters. On the real line, $f(x)=\max (0,1-|x|)$ is nonzero exactly on $(-1,1)$, but its support is the closed interval $[-1,1]$. A point where $f$ vanishes can still belong to the support if nonzero values occur arbitrarily near it.

For the discrete topology, every subset is closed, so the support is simply the set of inputs with nonzero values. This is the convention used when representing a finitely supported count function. A function has compact support when the displayed closed set is compact.

A vector-valued function is a function $F:X\to V$ taking values in a vector space. For example, $F(t)=(\cos t,\sin t)$ describes the unit circle in ${\mathbb{R}}^2$: one real number controls a two-component vector.

In a real or complex finite-dimensional vector space equipped with a norm, choosing a basis writes $F={\sum }_j{f}_j{e}_j$ in scalar-valued functions. Continuity and differentiability can then be checked in these coordinates, and $F^{\prime }={\sum }_j{f}_j^{\prime }{e}_j$ when the derivatives exist. Finitely many coordinates matter: coordinatewise convergence in an infinite-dimensional vector space need not imply norm convergence.