Analysis II | Part IB, 2002

(a) Let VV be a finite-dimensional real vector space, and let 11\left\|_{1}\right\|_{1} and 2\|\cdot\|_{2} be two norms on VV. Show that a function f:VRf: V \rightarrow \mathbb{R} is differentiable at a point aa in VV with respect to 1\|\cdot\|_{1} if and only if it is differentiable at aa with respect to 2\|\cdot\|_{2}, and that if this is so then the derivative f(a)f^{\prime}(a) of ff is independent of the norm used. [You may assume that all norms on a finite-dimensional vector space are equivalent.]

(b) Let V1,V2V_{1}, V_{2} and V3V_{3} be finite-dimensional normed real vector spaces with VjV_{j} having norm j,j=1,2,3\|\cdot\|_{j}, j=1,2,3, and let f:V1×V2V3f: V_{1} \times V_{2} \rightarrow V_{3} be a continuous bilinear mapping. Show that ff is differentiable at any point (a,b)(a, b) in V1×V2V_{1} \times V_{2}, and that f(a,b)(h,k)=f^{\prime}(a, b)(h, k)= f(h,b)+f(a,k).f(h, b)+f(a, k) . \quad [You may assume that (u12+v22)1/2\left(\|u\|_{1}^{2}+\|v\|_{2}^{2}\right)^{1 / 2} is a norm on V1×V2V_{1} \times V_{2}, and that {(x,y)V1×V2:x1=1,y2=1}\left\{(x, y) \in V_{1} \times V_{2}:\|x\|_{1}=1,\|y\|_{2}=1\right\} is compact.]

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