Boman's theorem


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  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

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        (&)(\Re \dashv \Im \dashv \&)

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          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous *

          \array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }



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          Boman's Theorem


          In considering certain types of generalized smooth spaces, one may try to describe the smooth structure on a space by specifying the smooth curves. Boman's Theorem shows that this is sufficient to describe the smooth structure on a smooth manifold.


          The following theorem appears (as Theorem 1) in the paper Boman 1967, where it is first proved, by Jan Boman:


          Let ff be a function from d\mathbb{R}^d to \mathbb{R}, and assume that the composed function fuf \circ u belongs to C (,)C^\infty(\mathbb{R},\mathbb{R}) for every uC (, d)u \in C^\infty(\mathbb{R}, \mathbb{R}^d). Then fC ( d,)f \in C^\infty(\mathbb{R}^d, \mathbb{R}).

          Here, d\mathbb{R}^d is a Cartesian space, and C (X,Y)C^\infty(X,Y) is the set of smooth maps from XX to YY.

          The theorem is quoted with a proof in Kriegl & Michor 1997 (Theorem 3.4).

          Less than smooth functions

          This theorem is for smooth functions, that is C C^\infty maps. A similar theorem could be stated for continuous functions, that is C 0C^0 maps. The situation is slightly less than ideal, however, for continuously differentiable functions, that is C 1C^1 maps, or more generally C pC^p maps for 0<p<0 \lt p \lt \infty.

          Boman 1967 has this as part of Theorem 2:


          Let ff be a function from d\mathbb{R}^d to \mathbb{R}, and assume that the composed function fuf \circ u belongs to C p(,)C^p(\mathbb{R},\mathbb{R}) for every uC (, d)u \in C^\infty(\mathbb{R}, \mathbb{R}^d). Then fC p1( d,)f \in C^{p-1}(\mathbb{R}^d, \mathbb{R}).

          Note that pp has become p1p - 1 in the conclusion. (Boman's full Theorem 2 gives stronger results involving Lipschitz conditions.)

          Boman's Theorem 8 gives the desired result if we use parametrized surfaces instead of curves:


          Let ff be a function from d\mathbb{R}^d to \mathbb{R}, and assume that the composed function fuf \circ u belongs to C p( 2,)C^p(\mathbb{R}^2,\mathbb{R}) for every uC ( 2, d)u \in C^\infty(\mathbb{R}^2, \mathbb{R}^d). Then fC p( d,)f \in C^p(\mathbb{R}^d, \mathbb{R}).

          Here we have 2\mathbb{R}^2 instead of \mathbb{R} as the domain of uu.

          Boman's Theorem 3 guarantees such counterexamples as

          f:x,yy 3x 2+y 2 f\colon x, y \mapsto \frac{y^3}{x^2 + y^2}

          (continuously extended so that f(0,0)=0f(0,0) = 0). Given any smooth —or even C 1C^1— curve u:t(g(t),h(t))u\colon t \mapsto (g(t), h(t)), it may be shown (by several tedious cases) that (fu)(f \circ u)' is continuous. Nevertheless, ff is not C 1C^1 at (0,0)(0,0). (The general pattern, expressed in Boman's Theorem 10, is to use a non-polynomial function that is homogeneous in degree pp and C pC^p except at 0\vec{0}. So long as d>1d \gt 1, such functions exist.) Additionally, (fu)(f \circ u)' exists even if uu is merely differentiable, but ff is not even differentiable at (0,0)(0,0).

          This does not contradict the well known theorem (often taken as a definition!) that a function is C 1C^1 already if only its partial derivatives are continuous; while the partial derivatives of ff may be expressed as derivatives of fuf \circ u for appropriate smooth uu (taken from a space of curves identifiable with d dd \mathbb{R}^d), the continuity of the partial derivatives requires not that (fu)(t)(f \circ u)'(t) be continuous in tt (although this will follow) but that it be continuous in uu.


          • Jan Boman, Differentiability of a function and of its compositions with functions of one variable, Math. Scand. 20 1967 249–268, MR237728 pdf

          • Andreas Kriegl, Peter W. Michor, The convenient setting of global analysis, Math. Surveys and Monographs 53, Amer. Math. Soc. 1997. x+618 pp. ISBN: 0-8218-0780-3 html MR1471480

          Last revised on April 10, 2016 at 18:47:50. See the history of this page for a list of all contributions to it.