Find the first derivative of this functional

[Dragonfall]

Homework Statement

For $$f\in C([0,1])$$, let $$N(f)=\int_0^1(f(t))^2dt$$. Show that $$N\in C^{\infty}$$ and calculate the first derivative.

Homework Equations

Can I use Leibniz's integral rule for this?

The Attempt at a Solution

If I just blindly plug in the formula, I get

$$dN/df=\int_0^12f(t)dt$$

 

[AKG]

You have some very strange notation. I assume:

$$C([0,1]) = \{ f : [0,1] \to \mathbb{R} | f\mbox{ is continuous}\}$$

That would make

$$C^{\infty} = \{ F : C([0,1]) \to \mathbb{R} | F\mbox{ is infinitely differentiable}\}$$

Is this what you mean? If so, how is the derivative of an elment of $C^{\infty}$ defined?

 

[Dragonfall]

Yes, that's what I mean. I don't understand your question. I suppose they can be defined in terms of its power series.

 

[AKG]

It was a pretty straightforward question. What is the definition of the derivative of an element of $C^{\infty}$?

 

[HallsofIvy]

AKG's point is that N is NOT a function from the real numbers to the real numbers. It is a function whose domain is a set of functions, and whose range is the real numbers. How are you defining the derivative of such a function?

 

[Dragonfall]

I actually don't know. I just know that in a general metric space, a function f is differentiable at one point y if it can be written as f(x)=f(y)+df(x-y)+p(x) where p is little-o norm and df(x-y) is a linear continuous function. I suspect df is the derivative, but I don't know how to compute it.

There is no mentioning of differentiation in my book (Elements of the Theory of Functions and Functional Analysis) and I might have made a mistake in my notes.

 

[HallsofIvy]

I actually don't know. I just know that in a general metric space, a function f is differentiable at one point y if it can be written as f(x)=f(y)+df(x-y)+p(x) where p is little-o norm and df(x-y) is a linear continuous function. I suspect df is the derivative, but I don't know how to compute it.

There is no mentioning of differentiation in my book (Elements of the Theory of Functions and Functional Analysis) and I might have made a mistake in my notes.

In a "general metric space", addition is not defined. What you are talking about is a topological vector space (the most general situation in which the derivative can be defined). Sets of functions do form topological vector spaces in which the topology is based on a metric but there are several different ones that are not equivalent. You haven't said which metric you are using.

 

[Dragonfall]

You haven't said which metric you are using.

Neither did the question.

Let's assume the supremum metric.

 

[AKG]

I actually don't know. I just know that in a general metric space, a function f is differentiable at one point y if it can be written as f(x)=f(y)+df(x-y)+p(x) where p is little-o norm and df(x-y) is a linear continuous function. I suspect df is the derivative, but I don't know how to compute it.

There is no mentioning of differentiation in my book (Elements of the Theory of Functions and Functional Analysis) and I might have made a mistake in my notes.

How is the little-o norm defined?

 

[Dragonfall]

Let V, W be vector spaces, let $$M\subset V$$ be open and let $$x_0\in M$$. Then $$f:M\rightarrow W$$ is little-o norm if $$\forall\epsilon>0\exists\delta>0\forall x(||x-x_0||_V<\delta\Rightarrow||f(x)-f(x_0)||_W<\epsilon||x-x_0||_V)$$.

 

[AKG]

Your definition of little-o norm depends on x₀, i.e. it could only be read to say that f is little-o norm at x₀. In your definition of differentiability, then, do you mean p to be little-o norm at some point, or at all points?

Also, the only way your definition makes sense is if you mean df to be linear continuous, not df(x-y).

 

[Dragonfall]

Is df(x-y) df evaluated at x-y or df times (x-y)? Also, the little-o norm is probably defined at the point which is differentiable, but I'm not sure. Is there a book where I might find the definition?

 

[AKG]

The definition that would make the most sense to me is that given f : V -> W, f is differentiable at some y in V iff there exists a function p : V -> W that is little-o norm at y and a function df(y) : V -> W that is continuous and linear such that for all x in V, f(x) = f(y) + (df(y))(x-y) + p(x). Note that if V = W = R, then a continuous linear function df(y) : V -> W just multiplies its argument by a constant. So if f'(y) is what we think of as being the derivative of f at y in the normal sense (i.e. it is just a number), then df(y)(z) = f'(y)z, where on the left we have the function df(y) acting on z, and on the right we have a number f'(y) multiplying with z.

I think N is differentiable, so I would proceed as follows: Pick an arbirary function f. Guess a function $dN(f) : C^{\infty} \to \mathbb{R}$ that's continuous and linear. Pick an arbitrary function g, and define $p : C^{\infty} \to \mathbb{R}$:

p(g) = N(g) - N(f) - (dN(f))(g-f)

Check that p so-defined is little-o norm at f. Guessing dN(f) is the tricky part, because as far as I know, you just have to make a reasonable guess, there's no smart way I know of of picking the right dN(f). This might be because this is the first time I'm seeing differentiation on an arbitrary normed vector space. If you pick the right dN(f), then showing p is little-o norm at f is just a first-year analysis sort of $\delta -\epsilon$ problem.

 

[Dragonfall]

That makes sense, but to be sure, can you recommend a book that would contain the definition? I'm not even sure which field this falls into.