7.2 Function Spaces

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Remark 7.2.1. C[a, b] has a natural inner product which is a generalisation

of the standard inner product on Rn. To see this, contemplate the geometrical

picture of the ‘dot’ product in figure 7.1

It should be clear that we could think of the functions from [a, b] to R as

vectors with a component for every t 2 [a, b] and if f, g are two such functions,

the equivalent of the dot product would be the infinite sum of f(t)g(t). It

should come as no surprise that we have:

< f, g > ,

Z b

a

f(t)g(t) dt

Proposition 7.2.1. < f, g > ,

R b

a f(t)g(t) dt is an inner product on C[a, b].

Proof:

< f,− >: C[a, b] −! R

7.2. FUNCTION SPACES 131

Figure 7.1: Inner Products

g 

Z b

a

f(t)g(t) dt

is clearly linear since

Z

(f(t)(g + h)(t) dt =

Z

f(t)g(t) dt +

Z

f(t)h(t) dt

and

8 _ 2 R,

Z

f(t)(_g)(t) dt = _

Z

f(t)g(t) dt

and 8f, g 2 C[a, b], < f, g >=< g, f > so <,> is bilinear.

Finally, it is trivial that

< f, f > _ 0

and

< f, f >= 0 )

Z b

a

(f(t))2 = 0

Now if 9 t 2 [a, b] : f(t) 6= 0, then (f(t))2 > 0 and there is a neighbourhood

of t for which f2 > 0 by continuity. Hence

R b

a f2 > 0. The contrapositive of

this is the proposition

< f, f >= 0 ) f = 0

_

132 CHAPTER 7. FOURIER THEORY

Remark 7.2.2. It follows that we have a derived metric on C[a, b] given by

d(f, g) =

sZ

(f(t) − g(t))2dt

and a norm given by

kfk =

sZ b

a

(f(t))2 dt

Remark 7.2.3. This is called the L2 norm on the space. There are others,

and other metrics. The idea of having different notions of distance on the

same set is a bit strange at first, but one gets used to it. Compare:

d(f, g) = sup

t2[a,b]

|f(t) − g(t)|

and

kfk = sup

t2[a,b]

|f(t)|

Remark 7.2.4. This gives another and different sense of the “distance”

between two functions and the “size” of a function. So don’t talk or think

of “the” distance between functions. These two are different and there are

others.

Remark 7.2.5. We need to use the idea of a distance between functions when

we are making precise the idea of a sequence of functions approximating to

a function. For example we might approximate some function by a sequence

of polynomials. It is important to be clear about the idea of a sequence

converging, but this makes sense in any metric space: which metric the

convergence occurs in is of some practical importance. For example, we

might be converging in the sense of the last metric, but wind up with a very

bad approximation to the derivative which got steadily worse as we converge

to the values of the function.

Remark 7.2.6. We can now say when two functions in C[a, b] are orthogonal;

it is when the inner product is zero.

Proposition 7.2.2. cos and sin are orthogonal in C[−_, _]

Proof:

Z _

−_

sin(t) cos(t) dt =

1

2

Z _

−_

sin(2t) dt =

−1

4

cos(2t)

__

−_

= 0

_

This can be strengthened:

7.2. FUNCTION SPACES 133

Proposition 7.2.3. sin(nt), cos(mt) are orthogonal on C[−_, _] for any integers

n,m.

Proof: If n = 0 then sin(nt) is just the zero function so the inner product

(integral) is certainly zero. If m = 0 then we have the constant function 1

and the claim is Z _

−_

sin(nt) = 0

but Z _

−_

sin(nt) =

−1

n

cos(nt)

__

−_

= 0

Finally if neither n nor m is zero we note that cos is an even function and

sin is an odd function, so the resulting function is odd, and for every positive

term there is a corresponding negative one in the integral, so the integral is

zero. _

Moreover:

Proposition 7.2.4. If n 6= m, sin(nt) and sin(mt) are orthogonal on C[−_, _]

Proof: Recall:

sin(A + B) = sinAcosB + sinB cosA

cos(A + B) = cosAcosB − sinAsinB

and bearing in mind that sin is odd and cos is even

2 sinAcosB = cos(A + B) + cos(A − B)

from which it follows that

sin(nt) sin(mt) =

1

2

[cos(n − m)t + cos(n + m)t]

So Z _

−_

sin(nt) sin(mt) =

1

2

Z _

−_

cos(n − m)t +

1

2

Z _

−_

cos(n + m)t

both terms being zero when n 6= m _

Finally:

Proposition 7.2.5. If n 6= m, cos(nt) and cos(mt) are orthogonal on

C[−_, _]

Proof: Left as an easy exercise. _

134 CHAPTER 7. FOURIER THEORY

Figure 7.2: Projection on an orthonormal set: P = U + V + W

Remark 7.2.7. Note that the underlying interval is crucial here. It all collapses

in C[−1, 1] for instance– although we could always change the functions

to be sin(n_x) and cos(n_x), and this would work.

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