赋范线性空间上微分学—映照可微性与高阶导数 谢锡麟 有 v(u)=t[f(xo+t(u+h2)-f(xo+tu)∈x(x;Y), 且θ(t)=v(h1)-v(0)∈Y.估计 )-t2f"(x0)(h2,h1)ly=|(h1)-v(0)-t2[r"(xo)(h2)(h1)y ≤tsup,|f(xo+t(mh1+h2)-f(xo+trh1)-tf"(o(h2)xy)hlx f(ro+t(rh1 + h2))=f(ro)+tf"(aro)(hi +h2)+o(tE(X;Y) f(co+trh1)=f(o)+trf"(ao(h1)+o(tEg(X; Y), 则有 le(t)t(ro)(h2, hily <t sup lo(t)lx(x r)lh1Ix=lot)le(x: r)lh1Ix, 即有 (t) =f(ao)(h2,h)∈Y 综上,按极限存在的唯一性,有 Co 考虑三阶导数,有 b>(xo)(h,bm,b)=「d2(d(0)(h,h,1) dx2(dr da)(ao)(h1,h2)(h3) (x0)(h2,h1)(h3) d rs(co)(h2, h1, 3), 另有 d 2 o)(h,h,h3)=3(mo)(hn)(h2,h3)=Dh(xo)(h2,h) =D/ (x0)(h2,h3 h[a2()(h3,h2) (x0)(h1,h3,h2) 反复利用上面的两个关系,可有 )(b(1),h(2,h(3),Va∈P3 对更高阶的情况,可做类似讨论赋范线性空间上微分学 赋范线性空间上微分学—— 映照可微性与高阶导数 谢锡麟 有 ψ ′ (u) = t [ f ′ (x0 + t(u + h2)) − f ′ (x0 + tu) ] ∈ L (X; Y ), 且 θ(t) = ψ(h1) − ψ(0) ∈ Y . 估计  θ(t) − t 2 f ′′(x0)(h2, h1)   Y =  ψ(h1) − ψ(0) − t 2 [ f ′′(x0)(h2) ] (h1)   Y 6 t sup τ∈(0,1)  f ′ (x0 + t(τh1 + h2)) − f ′ (x0 + tτh1) − tf′′(x0)(h2)   L (X;Y ) |h1|X. 由 f ′ (x0 + t(τh1 + h2)) = f ′ (x0) + tf′′(x0)(τh1 + h2) + o(t) ∈ L (X; Y ) f ′ (x0 + tτh1) = f ′ (x0) + tτf′′(x0)(h1) + o(t) ∈ L (X; Y ), 则有  θ(t) − t 2 f ′′(x0)(h2, h1)   Y 6 t sup τ∈(0,1) |o(t)|L (X;Y ) |h1|X = |o(t 2 )|L (X;Y ) |h1|X, 即有 ∃ lim t→0 θ(t) t 2 = f ′′(x0)(h2, h1) ∈ Y. 综上, 按极限存在的唯一性, 有 f ′′(x0)(h1, h2) = f ′′(x0)(h2, h1). 考虑三阶导数, 有 d 3f dx 3 (x0)(h1, h2, h3) = [ d 2 dx 2 ( df dx ) (x0) ] (h1, h2, h3) = [ d 2 dx 2 ( df dx ) (x0)(h1, h2) ] (h3) = [ d 2 dx 2 ( df dx ) (x0)(h2, h1) ] (h3) = d 3f dx 3 (x0)(h2, h1, h3), 另有 d 3f dx 3 (x0)(h1, h2, h3) = [ d 3f dx 3 (x0)(h1) ] (h2, h3) = [ Dh1 d 2f dx 2 (x0) ] (h2, h3) = Dh1 [ d 2f dx 2 (x0)(h2, h3) ] = Dh1 [ d 2f dx 2 (x0)(h3, h2) ] = d 3f dx 3 (x0)(h1, h3, h2). 反复利用上面的两个关系, 可有 d 3f dx 3 (x0)(h1, h2, h3) = d 3f dx 3 (x0)(hσ(1), hσ(2), hσ(3)), ∀ σ ∈ P3. 对更高阶的情况, 可做类似讨论. 12