There is a way to get the differentiability of a function from the existence and continuity of the Gateaux
derivatives. This is very convenient because these Gateaux derivatives are taken with respect to a one
dimensional variable. The following theorem is the main result.
Theorem 17.5.1Let X be a normed vector space having basis
{v1,⋅⋅⋅,vn}
and let Y be another normedvector space having basis
{w1,⋅⋅⋅,wm }
. Let U be an open set in X and let f : U → Y have the propertythat the Gateaux derivatives,
f (x +tvk)− f (x)
Dvkf (x) ≡ lit→m0----------------
t
which proves the continuity of Df because of the assumption the Gateaux derivatives are continuous.
■
This motivates the following definition of what it means for a function to be C^{1}.
Definition 17.5.2Let U be an open subset of a normed finite dimensional vector space, X and letf : U → Y another finite dimensional normed vector space. Then f is said to be C^{1}if there exists a basis forX,
{v1,⋅⋅⋅,vn}
such that the Gateaux derivatives,
Dvk f (x)
exist on U and are continuous.
Note that as a special case where X = ℝ^{n}, you could let the v_{k} = e_{k} and the condition would reduce to
nothing more than a statement that the partial derivatives
∂∂fxi
are all continuous.
Here is another definition of what it means for a function to be C^{1}.
Definition 17.5.3Let U be an open subset of a normed vector space, X and let f : U → Y anothernormed vector space. Then f is said to be C^{1}if f is differentiable and x → Df
(x)
is continuous asa map from U to ℒ
(X, Y)
.
Now the following major theorem states these two definitions are equivalent. This is obviously so in the
special case where X = ℝ^{n} and the special basis is the usual one because, as observed earlier, the matrix of
Df
(x)
is just the one which has for its columns the partial derivatives which are given to be
continuous.
Theorem 17.5.4Let U be an open subset of a normed finite dimensional vector space, X andlet f : U → Y another finitedimensional normed vector space. Then the two definitions above areequivalent.
Proof: It was shown in Theorem 17.5.1, the one about the continuity of the Gateaux derivatives
yielding differentiability, that Definition 17.5.2 implies 17.5.3. Suppose then that Definition 17.5.3 holds.
Then if v is any vector,
f (x+-tv)−-f (x) Df-(x)tv-+-o(tv)
ltim→0 t = ltim→0 t
o-(tv)
= Df (x)v+ ltim→0 t = Df (x)v
Thus D_{v}f
(x)
exists and equals Df
(x )
v. By continuity of x → Df
(x)
, this establishes continuity of
x → D_{v}f
(x)
and proves the theorem. ■
Note that the proof of the theorem also implies the following corollary.
Corollary 17.5.5Let U be an open subset of a normed finite dimensional vector space, X and letf : U → Y another finite dimensional normed vector space. Then if there is a basis of X,
{v1,⋅⋅⋅,vn}
such that the Gateaux derivatives, D_{vk}f
(x )
exist and are continuous. Then all Gateaux derivatives,D_{v}f
(x)
exist and are continuous for all v ∈ X.
From now on, whichever definition is more convenient will be used.