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Unformatted text preview: 7.5. HARMONIC FORMS 191 • The coderivative is a linear map δ : Λ p → Λ p- 1 defined by δ = *- 1 d * = (- 1) ( n- p + 1)( p- 1) * d * . • The exterior derivative and the coderivative satisfy the important conditions d 2 = δ 2 = . • Problem . Show that in local coordinates the coderivative of a p-form α is the ( p- 1)-form δα with components ( δα ) i 1 ... i p- 1 = g i 1 j 2 ··· g i p- 1 j p 1 p | g | ∂ j p | g | g jk 1 g j 2 k 2 ··· g j p k p α k 1 k 2 ... k p • Now we define the L 2-inner product of p-forms by ( α, β ) L 2 = Z M α ∧ * β = Z M h α, β i vol , and the L 2-norm || α || L 2 = p ( α, α ) L 2 . • This makes the space C ∞ ( Λ p ( M )) of smooth p-forms an inner-product vec- tor space. • The completion of C ∞ ( Λ p ( M )) in the L 2-norm gives the Hilbert space L 2 ( Λ p ( M )) of square-integrable p-forms. • Let A : H → H be an operator on a Hilbert space H . The adjoint of the operator A with respect to the inner product of the space...
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