Calculus on Normed Vector Spaces by Rodney Coleman

Author:Rodney Coleman
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

5.4 Appendix: Homogeneous Polynomials

Suppose that E and F are vector spaces (not necessarily normed). If ϕ is a mapping from E into F and there is a k-linear mapping from E into F such that

for x ε E, then ϕ is said to be a k-homogeneous polynomial. The zero mapping is k-homogeneous for any k. However, for any other homogeneous polynomial, the number k is unique, because

for . We call this k the degree of ϕ. By convention we say that the zero mapping is of degree − ∞ and that a nonzero constant mapping from E into F is a homogeneous polynomial of degree 0.

Remark.

If and k = 2, then we call ϕ a quadratic form.

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