Lecture 1. Notion of Derivative
by Nirakar Neo
We shall always assume that are Banach spaces over the complex field . The real number field is denoted by . We shall now explore differential calculus in Banach spaces. It will be instructive to try to understand the theorems in well known Banach spaces like (over ) or (over ) or (over ). In general there are two notions of derivatives – the Gateaux derivative and the Frechet derivative, where the latter is stronger in the sense that Frechet differentiability implies Gateaux differentiability.
Definition: Let , where are Banach space. For , define
provided the limit exists. Then is called the directional derivative of at in the direction of . If this limit exists for every and if the function is linear and continuous in , then we say that the function is Gateaux differentiable and the function defined by
is called the Gateaux derivative at . Thus the linearity and continuity condition means
[An example to show that the map is not always linear, and even if exists, it may not be continuous at .]
Definition: Let , where are Banach space. Let be open subset. We say that is Frechet differentiable at , if there exists a bounded linear map such that
Some remarks should be in order, which we leave as simple exercises. The Frechet derivative is unique, and Frechet differentiability implies continuity. If exists, then also exists, and we have
Usually when we say is differentiable, we shall mean Frechet differentiable. [An example for which the Gateaux derivative exists but Frechet derivative fails to exist.]
Our next step is to prove the mean value theorem.
Theorem: Let , and suppose exists for all , and is a continuous map of , then
Proof: Let . Define is differentiable, since it is the composition of two differentiable function. We have . Using the fundamental theorem of calculus for real valued functions, we get
Also, , so altogether we get,
Since this holds for all , we get the desired result.
Notice further that
This is referred to as the mean value theorem in higher dimensions. The usual mean value theorem does not hold in higher dimensions as is seen by this example. Define for . Then is differentiable, and , but there is no such for which is zero.
Our next step is to generalize the chain rule. However we shall not prove the rule using strong hypotheses that both functions in a composition are Frechet differentiable. Instead we have –
Theorem: Let . Suppose is Frechet differentiable at and is having directional derivative in the direction at , then has a directional derivative at , and
Before plunging into a proof, let us have an informal discussion. On the right hand side of the above equation, we have a linear transformation acting on an element of Y – . Let us see what we have got. has directional derivative at in the direction of . This translates into
that is given there exists such that if we have
is Frechet differentiable at , so
where the right hand side goes to zero as . We may take
which certainly goes to zero at . If we plug this in the equation of , we’ll get
To complete the proof we merely have to show that the last two terms go to zero. This we shall prove formally in the next post.