View source for Homogeneous operator

A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047680/h0476801.png" /> of a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047680/h0476802.png" /> into a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047680/h0476803.png" /> such that there exists a symmetric [[Multilinear mapping|multilinear mapping]]

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with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047680/h0476805.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047680/h0476806.png" /> of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047680/h0476807.png" /> is called the degree of the homogeneous operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047680/h0476808.png" />. A linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047680/h0476809.png" /> is a homogeneous operator of degree 1 (usually just called homogeneous). One writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047680/h04768010.png" /> instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047680/h04768011.png" /> for short, meaning by this the element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047680/h04768012.png" /> with all coordinates equal, but not a power of an element — a concept that is not defined in an arbitrary vector space. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047680/h04768013.png" /> is a homogeneous operator of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047680/h04768014.png" />, then

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More generally:

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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047680/h04768018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047680/h04768019.png" /> are normed vector spaces, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047680/h04768020.png" /> is continuous if and only if it is bounded, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047680/h04768021.png" /> is continuous at zero it is continuous on the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047680/h04768022.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Lyusternik,   V.I. Sobolev,   "Elements of functional analysis" , Hindushtan Publ. Comp.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  H. Cartan,   "Calcul différentiel" , Hermann  (1967)</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  H. Cartan,   "Differential forms" , Kershaw  (1983)  (Translated from French)</TD></TR></table>