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By Jean-François Treves

Publication via Treves, Jean-François

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E,, fi, . . fs ) be r + s linearly inde­ pendent vectors in V, such that j = 1 , . . , r, k = 1 , . . , s. There exist 2n - (r + s) vectors e, +i , . . s + i. . , fn, such that (e i , . . , em fi. . , fn ) is a symplectic basis of V. PROOF. , W2 ) denote the linear span of e i , . . , of fi. . . fs ). Both W1 and W2 are isotropic, because of ( 1 . 14). 2.

This is true, in parti­ cular , if we assume that n is bounded and that A (x, t, g) extends as a C00 function to an open neighborhood of the closure of n, and if T > 0 is sufficiently small. The solution w is always real-valued, of course, and because of the uniqueness and of the fact that the initial datum x · 8 is homogeneous of degree one in 8, w (x, t, 8) is positive-homogeneous of degree one in 8. 20)

Let (e i, . . , e,, fi, . . fs ) be r + s linearly inde­ pendent vectors in V, such that j = 1 , . . , r, k = 1 , . . , s. There exist 2n - (r + s) vectors e, +i , . . s + i. . , fn, such that (e i , . . , em fi. . , fn ) is a symplectic basis of V. PROOF. , W2 ) denote the linear span of e i , . . , of fi. . . fs ). Both W1 and W2 are isotropic, because of ( 1 . 14). 2.

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