By P.R. Halmos

From the Preface: "This booklet used to be written for the lively reader. the 1st half comprises difficulties, often preceded via definitions and motivation, and infrequently through corollaries and old remarks... the second one half, a really brief one, includes hints... The 3rd half, the longest, contains suggestions: proofs, solutions, or contructions, looking on the character of the problem....

This isn't really an creation to Hilbert area concept. a few wisdom of that topic is a prerequisite: at the least, a examine of the weather of Hilbert area concept should still continue at the same time with the studying of this book."

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**Extra info for A Hilbert Space Problem Book**

**Example text**

Indeed XH (H) = dH(XH ) = ω(XH , XH ) = 0. 2. Let (M1 , ω1 ) and (M2 , ω2 ) be two symplectic manifolds. A diffeomorphism f : M1 → M2 is symplectic if and only if f∗ (XH◦f ) = XH for every open set U ⊂ M2 and smooth function H : U → R. Proof. The condition XH (f (p)) = f∗p (XH◦f (p)) for every p ∈ f −1 (U ) is equivalent to ω2 (f (p))(XH (f (p)), f∗p (v)) = ω2 (f (p))(f∗p (XH◦f (p)), f∗p (v)) for every v ∈ Tp M , since ω2 is non-degenerate and f is a diffeomorphism. Equivalently, dH(f (p))(f∗p (v)) = (f ∗ ω2 )(p)(XH◦f (p), v) or iXH◦f ω1 = d(H ◦ f ) = f ∗ (dH) = iXH◦f (f ∗ ω2 ) on f −1 (U ).

The condition XH (f (p)) = f∗p (XH◦f (p)) for every p ∈ f −1 (U ) is equivalent to ω2 (f (p))(XH (f (p)), f∗p (v)) = ω2 (f (p))(f∗p (XH◦f (p)), f∗p (v)) for every v ∈ Tp M , since ω2 is non-degenerate and f is a diffeomorphism. Equivalently, dH(f (p))(f∗p (v)) = (f ∗ ω2 )(p)(XH◦f (p), v) or iXH◦f ω1 = d(H ◦ f ) = f ∗ (dH) = iXH◦f (f ∗ ω2 ) on f −1 (U ). This is true, if f is symplectic. Conversely, if this holds, then for every p ∈ M1 and u, v ∈ Tp M1 there exists an open neighbourhood U of f (p) in M2 and a smooth function H : U → R such that u = XH◦f (p).

Therefore, df = 0 on M . This shows that the structural matrix is invertible. If M is connected, the Poisson structure is also non-degenerate. For the converse, let M be a Poisson manifold, such that the structural matrix W is everywhere invertible. For f ∈ C ∞ (M ) put Xf = adf . We define ω(Xf , Xg ) = {f, g} = W (df, dg) = df (Xg ). Since Tp∗ M is generated by {(dg)(p) : g ∈ C ∞ (M )} and W is invertible, it follows that ω is a non-degenerate 2-form and it remains to show that ω is closed. For this, we observe first that [Xf , Xg ](h) = Xf (Xg (h)) − Xg (Xf (h)) = Xf ({h, g}) − Xg ({h, f }) = {{h, g}, f } − {{h, f }, g} = −{h, {f, g}} = −X{f,g} (h) for every h ∈ C ∞ (M ).