New PDF release: Functional Operators.The geometry of orthogonal spaces

By John von Neumann

Measures and integrals

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Additional info for Functional Operators.The geometry of orthogonal spaces

Example text

A basic result in commutative algebra tells us, that any prime ideal of height one in a UFD is principal. Therefore, p = (/) for some irreducible / € Oc,o- In other words, any irreducible analytic germ of codimension one is defined by one irreducible holomorphic function. More generally, any analytic germ of codimension one is the zero set of a single holomorphic function. In the theory of functions of one variable a meromorphic function / on an open subset U C C is a holomorphic function defined on the complement of a discrete set of points S C U such that / has poles of finite order in all points of S.

E. Aa is uniquely determined by the condition (Aa, j3) = (a, L/3) for all /? € / \ V*. The C-linear extension /\* V£ —> /\* V^? of the dual Lefschetz operator will also be denoted by A. 3). Thus, the Hodge ^-operator is well-defined. Using an orthonormal basis xi,j/i = I(x\),... ,xn,yn = I(xn) as above, a straightforward calculation yields n\ • ujn = vol, where ui is the associated fundamental form. 9 for a far reaching generalization of this. e. A{/\ V*) C /\ ~ V*. Moreover, one has A = *~1 o L o *.