Variance Value Theorem 2: A Foray Into Functional Analysis

The Variance-Value Theorem: A foray into functional analysis Ogan Gurel 2 January 1993 Recall the Variance-Value Theorem which states that for any function f ( t ) continuous and differentiable over an interval [ a , b ] :

L f ( b ) f ( a ) O M P M N b a P Q 2

2 There exists a t * [a, b], such that [ f  ( t *)] 

It seemed strange that despite the deep ideas embodied in this result, one wasn't able to find any formulation of this theorem in any of the elementary calculus or introductory real analysis texts. I was curious to know why in these texts the mean-value theorem took precedence over the variance value theorem. A practical answer is that the mean-value theorem is more useful, especially for the purposes of using calculus to approximate functions. With the mean-value theorem one can thus formulate a linear approximation to a function based on its derivative: f (t ) f  ( a )[ t a ]

which serves as a prelude to Taylor's theorem and generalized power series expansions. 2

Nevertheless I was still convinced of the importance of [ f  ( t )] . I had originally interpreted this as the variance. Upon further reading, however, I've come across the concept of the norm of a function G ( x ) , which is a preliminary to the Banach spaces of functional analysis: On page 155 (under problem 4.8.10) of Avner Friedman's Foundations of Modern Analysis, Dover, 1982, one can find the following inequality: Let X be a real normed linear space, and let u ( t ) be continuous and strongly differentiable in ( a , b ). Then for any a b ,

u ( b ) u ( a ) (  ) sup  t 

du ( t ) dt

which is equivalent, though more strongly formulated, to the "variancevalue" theorem. It turns out that the concept of the norm of a function underlies functional analysis. Thus, my original curiosity was answered in two ways. First the variance value theorem can be found in more advanced discussions of analysis, and second, the concept of a norm of a function is important in further developments. In this sense then, these thoughts were an early foray into functional analysis. In conceptual terms, the variance value theorem theorem sets a minimum on the norm of a derivative, namely that it cannot be less than the norm of

a straight line.

For the sake of simplicity, let us define the function

over the unit interval [ 0 , 1] , rewrite f ( t ) u ( t ) and f  (t ) 

du ( t )

and rearrange.

dt

The variance value theorem is thus: f (t )

sup



f (1) f ( 0 )

 t 

But the next question arises: is there a maximum to the norm of a derivative? Intuitively, one would think that there should be no such maximum since there really should be no limit to how fast a function may change. Consider the function diagrammed below: f(t)

f(b)

G

f(a)

F

E

a

b

t

For the curve representing the function, there must be a maximum to the arc length over the interval [ a , b ] . In other words, for the curve to represent a function, a vertical line can only cut the curve once,: the curve cannot "curl" back over itself. From the above diagram, the maximum arc length over this interval would is the sum of the lengths, EF and FG, given by the formula: rmax  b a f ( b ) f ( a )

The formula for arc length gives the length of the curve over the interval (for a function differentiable over the interval) and thus this must be less than rmax .

z b

2

1 [ f  ( t )] dt  rmax  b a f ( b ) f ( a )

rab 

a

For simplicity, we take the interval [ 0 , 1] , so that:

z 1

2

1 [ f  ( t )] dt

0

 1  f (1) f ( 0 )

Which by the following manipulations,

z z z 1

2

1 [ f  ( t )] dt

1



f (1) f ( 0 )

0

1

2

1 [ f  ( t )]  1 dt



f (1) f ( 0 )

0

1

[f ( t )]

2

dt



f (1) f ( 0 )

0

But, by the definition of the norm of a function:

z 1

G( x )



2

[ G ( x )] dx

0

we get the following inequality: f (t )



f (1) f ( 0 )

This equation tells us that the norm of the derivative must be less than or equal to the total change of the function. Contrast this with the previous result of the variance value theorem: f (t )

sup



f (1) f ( 0 )

 t 

This equation states paradoxically that the norm of the derivative must be greater than or equal to the total change of the function Since both conditions are true, we can thus state: f (t )



f (1) f ( 0 )

In other words, the norm of the derivative is exactly equal to the total change of the function. And using the definition of the norm, we get:

z 1

f ( t ) dt 

f (1) f ( 0 )

0

and for an arbitrary interval [ a , b ] :

z b

f ( t ) dt 

f ( b ) f ( a )

a

which is, of course, the Fundamental Theorem of the Calculus. Hence by considering the absolute limits of a function (its minimum and maximum possible lengths) defined over a metric space we converge to the Fundamental Theorem. This is a remarkable result. We analyzed the minimum length by appealing to differentiation, and likewise the maximum length via integration and found out that they were in a strange sense

the same so that miraculously all differentiable functions straddle right in between these limits. As a bonus we find that integration is the reverse of differentiation. Put in these terms, the Fundamental Theorem of the Calculus can be more deeply thought of as the Fundamental Theorem of Functional Analysis.