7.7. HIGHER ORDER DERIVATIVES 193

Thus Dvf (x) exists and equals Df (x)v. By continuity of x→ Df (x) , this establishescontinuity of x→ Dvf (x) and proves the theorem. ■

Note that the proof of the theorem also implies the following corollary.

Corollary 7.6.5 Let U be an open subset of a normed finite dimensional vector space, Xand let f :U→Y another finite dimensional normed vector space. Then if there is a basis ofX ,{v1, · · · ,vn} such that the Gateaux derivatives, Dvkf (x) exist and are continuous, thenall Gateaux derivatives, Dvf (x) exist and are continuous for all v ∈X. Also Df (x)(v) =Dvf (x).

From now on, whichever definition is more convenient will be used.

7.7 Higher Order DerivativesIf f : U ⊆ X → Y for U an open set, then x→ Df (x) is a mapping from U to L (X ,Y ), anormed vector space. Therefore, it makes perfect sense to ask whether this function is alsodifferentiable.

Definition 7.7.1 The following is the definition of the second derivative. D2f (x)≡D(Df (x)) .

Thus, Df (x+v)−Df (x) = D2f (x)v+o(v) .This implies

D2f (x) ∈L (X ,L (X ,Y )) , D2f (x)(u)(v) ∈ Y,

and the map (u,v)→D2f (x)(u)(v) is a bilinear map having values in Y . In other words,the two functions,

u→ D2f (x)(u)(v) , v→ D2f (x)(u)(v)

are both linear.The same pattern applies to taking higher order derivatives. For example, D3f (x) ≡

D(D2f (x)

)and D3f (x) may be considered as a trilinear map having values in Y . In

general Dkf (x) may be considered a k linear map. This means

(u1, · · · ,uk)→ Dkf (x)(u1) · · ·(uk)

has the property u j→ Dkf (x)(u1) · · ·(u j) · · ·(uk) is linear.Also, instead of writing D2f (x)(u)(v) , or D3f (x)(u)(v)(w) the following notation

is often used.D2f (x)(u,v) or D3f (x)(u,v,w)

with similar conventions for higher derivatives than 3. Another convention which is oftenused is the notation Dkf (x)vk instead of Dkf (x)(v, · · · ,v) .

Note that for every k, Dkf maps U to a normed vector space. As mentioned above,Df (x) has values in L (X ,Y ) ,D2f (x) has values in L (X ,L (X ,Y )) , etc. Thus it makessense to consider whether Dkf is continuous. This is described in the following definition.

Definition 7.7.2 Let U be an open subset of X , a normed vector space, and letf : U → Y. Then f is Ck (U) if f and its first k derivatives are all continuous. Also,Dkf (x) when it exists can be considered a Y valued multi-linear function. Sometimesthese are called tensors in case f has scalar values.