Homogeneous operator

A mapping $ A $ of a vector space $ X $ into a vector space $ Y $ such that there exists a symmetric multilinear mapping

$$ B : X \times \dots \times X \rightarrow Y $$

with $ B ( x \dots x ) = A ( x) $. The number $ n $ of variables $ x $ is called the degree of the homogeneous operator $ A $. A linear operator $ L : X \rightarrow Y $ is a homogeneous operator of degree 1 (usually just called homogeneous). One writes $ x ^ {n} $ instead of $ ( x \dots x ) $ for short, meaning by this the element of $ X \times \dots \times X $ with all coordinates equal, but not a power of an element — a concept that is not defined in an arbitrary vector space. If $ A $ is a homogeneous operator of degree $ n $, then

$$ A ( tx ) = t ^ {n} A ( x) . $$

More generally:

$$ A ( t _ {1} x _ {1} + \dots + t _ {k} x _ {k} ) = $$

$$ = \ \sum _ {\begin{array}{c} n _ {1} + \dots + n _ {k} = n \\ n _ {i} \geq 0 \end{array} } \frac{n ! }{n _ {1} ! \dots n _ {k} ! } t _ {1} ^ {n _ {1} } \dots t _ {k} ^ {n _ {k} } B ( x _ {1} ^ {n _ {1} } \dots x _ {k} ^ {n _ {k} } ) . $$

If $ X $ and $ Y $ are normed vector spaces, then $ A $ is continuous if and only if it is bounded, and if $ A $ is continuous at zero it is continuous on the whole of $ X $.

References

[1]	L.A. Lyusternik, V.I. Sobolev, "Elements of functional analysis" , Hindushtan Publ. Comp. (1974) (Translated from Russian)
[2a]	H. Cartan, "Calcul différentiel" , Hermann (1967)
[2b]	H. Cartan, "Differential forms" , Kershaw (1983) (Translated from French)