Functions across vector spaces
\((y_{1} \dots) = f(x_{1}, \dots): C^{n} \to C^{m}\).
Called Operators on vector space by viewing vectors as functions.
If V is a Euclidian space, you have a vector field.
Also see functions over convex and affine spaces.
Functional sequence view
An important way to view a vector function is as a sequence of vector functionals. Thus, many properties of vector functions can be easily understood in terms of the properties of functionals. For example, the differential function is defined by combining differential functions corresponding to the constituent functionals.
Properties
Also see analysis of functions over fields.
Topological properties
For properties which arise when viewed as a metric space, see topology ref. Continuity properties of functions carry over from the continuity properties of functionals.
Linearity
See subsection on linear functions.
Bilinearity
Bilinear function: \(f(a+b, c) = f(a, c) + f(b, c)\): if ye hold one ip fixed, ye get linearity wrt other var.
Eg: \(f(x,y) = xy\).
Similarly, multilinearity is defined.
Distributive law
Just like \(xy\), the distributive law holds for all multilinear functions.
Proof
Easy to see for bilinear function. The multilinear case then follows by induction.
Generalized convexity
Consider inequalities defined by a pointed cone. If \(f(tx + (1-t)y) \preceq tf(x) + (1-t)f(y)\), \(f\) is convex. Many properties analogous to convexity of functionals, similarly proved. Epigraph of \(f\) is convex. Sublevel-sets of \(f\) are also convex.