laitimes

Integral properties of bounded linear functionals

Here, according to the explanation of a teacher on station B, the integral properties of bounded linear functionals are proved.

Integral properties of bounded linear functionals
Integral properties of bounded linear functionals
Integral properties of bounded linear functionals
Integral properties of bounded linear functionals

The assumption here is to treat Xs as a function with an independent variable s, as shown in the figure below:

Integral properties of bounded linear functionals

That is, Xs is not 0 only in the infinitesimal region of [a,s], and all of them are 0 in the interval [s,b].

The purpose of representing Xs in this way is equivalent to thinking that the function xs corresponding to the bounded linear functional f(xs) is only non-0 at a definite point.

Integral properties of bounded linear functionals
Integral properties of bounded linear functionals

where δk is the length of the cell consisting of [sk,tk].

The above proof shows that when f is a continuous linear functional, f(xs) is an absolute continuous function.

Since it is absolutely continuous, g(s) is differentiable, from which the following assumptions can be made:

Integral properties of bounded linear functionals

The last equation holds because Xs is 1 in the infinitesimal region of [a,s] and 0 in the interval [s,b].

The above proof shows that for the bounded linear functional f(x), since x can be approximated by the step function, it is absolutely continuous, thus satisfying Newton's Leibniz formula.

Integral properties of bounded linear functionals

This is because the stepped function can be expressed as

Integral properties of bounded linear functionals

The purpose of all the above proofs is to prove, for the bounded linear functional f(x), by proving its absolutely continuous properties, to obtain:

Integral properties of bounded linear functionals
Integral properties of bounded linear functionals