Errata for the book: The convenient setting of Global Analysis

Misprints and additions in the book: The convenient setting of Global Analysis, by Andreas Kriegl and Peter W. Michor

June, 2000. August, 2008. March, 2015.

Please, send any other misprints to <Peter.Michor@univie.ac.at>

page_{line number from below}
page^{line number from above}

• x_1: remove the DOS-gremlin
• 1_{16}: on a finite dimensional compact smooth manifold
• 27: In second display from bottom: \sum_{i=0}
• 56_12 Omit the last sentence.
• 66 In 5.26 Theorem: ... Then $C^\infty(U,F)$ is convenient and satisfies ...
• 106^4 Wording of Definition
• 116^4: Italian
• 116_5: The Portuguese
• 127: In 12.7 Lemma: replace locally bounded' by bounded on compact sets'. Some changes in the proof required. (Noted by Helge Glöckner)
• 187_8 not measurable
• 332^7: one each' is too much
• 369^{11}: is to assume that
• 374^1: Proof of (2).
• 374_5: Proof of (3).
• 429^{14}: on a finite dimensional compact manifold
• 450: The proof of Theorem 42.21 works only for a finite dimensional structure group. (Noted by Christoph Wockel)
• 451^{16}: u=(\exp|_V)^{-1}:U\to V
• 453_8: TM as last entry in the display
• 456 In 43.2 Example ... that $\Diff(M)$ contains a smooth curve through $\Id_M$ whose points (sauf $\Id_M$) are free generators of an arcwise connected free subgroup which meets the image of $\Exp$ only at the identity.
• 474_{15}: submani\-fold
• 475: More details in the proof of theorem 44.1 added.
• 475_2: C^\infty_c
• 476^{20}: $N$ twice,
• 476: The statement of 44.2 is enlarged and proofs have been included (10 pages more).
• 498^6: De Vrie{\ss} -> De Vries (often!)
• 499_2: +\frac{k_x(h\ell)_x}{f_x^2}
• 499_1 +2f_xhk_x\ell_x -2f_xh_xk\ell_x\Bigr)
• 501_1: = -\ad([X,Y]_\g)^\top.
• 503_1: +\tfrac14[\al(y),\al(u)]u
• 508_2: \align b_{tt} &= \int_{S^1}(-y_{txxx}u + y_{xxx}(3u_xu+au_{xxx}))dx \tag{\nmb|{2}'} \endalign
• 523^6 Then $\ell'$ is a Lie algebra homomorphism.
• 523 In 48.2 replace the second paragraph by the following which is needed in the proof of 48.8 Theorem (Otherwise it is wrong):
A $2$-form $\si\in\Om^2(M)$ is called a {\it weak symplectic structure}\index{weak symplectic structure} on $M$ if the following three conditions holds: \begin{enumerate} \item[(\nmb:{1})] $\si$ is closed, $d\si=0$. \item[(\nmb:{2})] The associated vector bundle homomorphism $\check\si: TM \to T^*M$ is injective. \item[(\nmb:{3})] The gradient of $\si$ with respect to itself exists and is smooth; this can be expressed most easily in charts, so let $M$ be open in a convenient vector space $E$. Then for $x\in M$ and $X,Y,Z\in T_xM=E$ we have $d\si(x)(X)(Y,Z)= \si(\Om_x(Y,Z),X) = \si(\tilde\Om_x(X,Y),Z)$ for smooth $\Om,\tilde\Om: M\x E\x E \to E$ which are bilinear in $E\x E$. \end{enumerate}
• 525 In 48.8 Theorem insert:
We equip $C^\infty_\si(M,\mathbb R)$ with the initial structure with respect to the the two following mappings: $$C^\infty_\si(M,\mathbb R) \East{\subset}{} C^\infty(M,\mathbb R),\qquad C^\infty_\si(M,\mathbb R) \East{\on{grad}^\si}{} \X(M).$$ Then the Poisson bracket is bounded bilinear on $C^\infty_\si(M,\mathbb R)$.
before: We have the following long exact sequence of Lie algebras
• 526 Many changes in the proof of the theorem.
• 527_{16}: remove er' and a second = sign
• 549^{16}: Omit sentence "All these norms ..."
• 609^{10}: Grundz\"uge