The discussion revolves around determining a constant \( c \) in relation to a function \( f \) that has a bounded derivative on the interval [0,1] with the condition that \( f(0)=0 \). The original poster questions whether \( c \) can be defined as \( c = \frac{2 ||f||_{\infty}}{||f||_E} \) and considers the case when \( f \) is zero everywhere.
Participants are actively engaging with the problem, providing hints and exploring various interpretations of the relationship between \( f \) and its derivative. Some guidance has been offered regarding the integral representation of \( f(x) \) in terms of \( f'(x) \), but no consensus has been reached on the implications of these relationships.
There is an emphasis on understanding the properties of functions within the context of Banach spaces, and participants express uncertainty about the connections between derivatives and functions in this framework. The original poster also notes a lack of reference material to clarify these concepts.
Yes, you're right!Dick said:The point to c being a constant is that it's independent of f, right?
Try and read the question eliminating the 'norm' words. You have a function f(x) that has a bounded derivative on [0,1] and f(0)=0. Can you get a bound for f(x) in terms of the bound on the derivative?
Dick said:If you are studying Banach spaces, you must know more about derivatives than just the definition. If you know f'(x) how can you find f(x)?
Take the absolute value of both sides of this, and then work from there.dirk_mec1 said:[tex]f(x) = \int_0^x f'(t)\ \mbox{d}t[/tex]
dirk_mec1 said:Do you mean like this?
[tex]|f(x)| = |\int_0^x f'(t)\ \mbox{d}t| \leq \int_0^x |f'(t)|\ \mbox{d}t \leq \int_0^x |f'(t)|_{\infty} \ \mbox{d}t \leq \int_0^1 M \mbox{d}t = M[/tex]
Is this correct?