赋范线性空间上微分学—映照可微性与高阶导数 谢锡麟 对于p=2的情况,有 d-1 d(x)(h1,h2)= (x)(h1)(h2)=|Dn (x)(h2) d (a+Anhi) df lim tr(h2,此处入n→0mn→+∞x) (x+Mnh1)(h2)-x(x)(/h2) lim limh2f(c+\b Dy= Dh,( Dh, f)(a)= Dh1 o Dh2 f(a) 对于p=3的情况,有 d 2 dr(z)(hi, a2, h3)=dr(a)(h1)(h2, h3)=Dh( dz2 )(a)(h2, h3) (a +Anhi) d2f (x) (a + Anhi)(h2) (x)(h (h3) d d dr (+ Anh)-Di lin D2(d)(x+h)3)-Dm2(a)(a3 =lim D (a + Anhi)-Dh2 o Dha f(ar) 另外,也可以利用p=2的结论,即 (x)(h1,h2,h3) (x)(h1)(h2,hy)=/2/0 ()(h21) d 2 m女2(x+mh1) (h2,h3) lim de2(a+Anh)(h2, h3) (x)(h2,h3) Dha o Dhs f(a+Anh1)-Dn2 o Dha f( Dha o Dha f(a)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 对于 p = 2 的情况, 有 d 2f dx 2 (x)(h1, h2) = [ d 2f dx 2 (x)(h1) ] (h2) = [ Dh1 ( df dx ) (x) ] (h2) =    lim df dx (x + λnh1) − df dx (x) λn    (h2), 此处λn → 0(n → +∞) = lim df dx (x + λnh1)(h2) − df dx (x)(h2) λn = lim Dh2 f(x + λnh1) − Dh2 f(x) λn = Dh1 (Dh2 f)(x) = Dh1 ◦ Dh2 f(x); 对于 p = 3 的情况, 有 d 3f dx 3 (x)(h1, h2, h3) = [ d 3f dx 3 (x)(h1) ] (h2, h3) = [ Dh1 ( d 2f dx 2 ) (x) ] (h2, h3) =     lim d 2f dx 2 (x + λnh1) − d 2f dx 2 (x) λn     (h2, h3) =     lim d 2f dx 2 (x + λnh1)(h2) − d 2f dx 2 (x)(h2) λn     (h3) =     lim Dh2 ( df dx ) (x + λnh1) − Dh2 ( df dx ) (x) λn     (h3) = lim Dh2 ( df dx ) (x + λnh1)(h3) − Dh2 ( df dx ) (x)(h3) λn = lim Dh2 ◦ Dh3 f(x + λnh1) − Dh2 ◦ Dh3 f(x) λn = Dh1 ◦ Dh2 ◦ Dh3 f(x). 另外, 也可以利用 p = 2 的结论, 即 d 3f dx 3 (x)(h1, h2, h3) = [ d 3f dx 3 (x)(h1) ] (h2, h3) = [ Dh1 ( d 2f dx 2 ) (x) ] (h2, h3) =     lim d 2f dx 2 (x + λnh1) − d 2f dx 2 (x) λn     (h2, h3) = lim d 2f dx 2 (x + λnh1)(h2, h3) − d 2f dx 2 (x)(h2, h3) λn = lim Dh2 ◦ Dh3 f(x + λnh1) − Dh2 ◦ Dh3 f(x) λn = Dh1 ◦ Dh2 ◦ Dh3 f(x). 10