_{j=1}^{n}, generated in this way are therefore an orthonormal basis because each
vector has unit length. ■
The process by which these vectors were generated is called the Gram Schmidt
process.
The following corollary is obtained from the above process.
Corollary 11.2.2Let X be a finite dimensional inner product space of dimension n whosebasis is
{u1,⋅⋅⋅,uk,xk+1,⋅⋅⋅,xn}
. Then if
{u1,⋅⋅⋅,uk}
is orthonormal, then the GramSchmidt process applied to the given list of vectors in order leaves
{u1,⋅⋅⋅,uk}
unchanged.
Lemma 11.2.3Suppose
{uj}
_{j=1}^{n}is an orthonormal basis for an inner product space X. Thenfor all x ∈ X,
∑n
x = (x,uj)uj.
j=1
Proof:Since
{u }
j
_{j=1}^{n} is a basis, there exist unique scalars
{α }
i
such that
∑n
x = αjuj
j=1
It only remains to identify α_{k}. From the properties of the inner product,
∑n n∑
(x,uk) = αj (uj,uk) = αjδjk = αk ■
j=1 j=1
The following theorem is of fundamental importance. First note that a subspace of an
inner product space is also an inner product space because you can use the same inner
product.
Theorem 11.2.4Let M be a finite dimensional subspace of X,an inner product space and let
{ei}
_{i=1}^{m}be an orthonormal basis for M. Then if y ∈ X and w ∈ M,
2 { 2 }
|y − w| = inf |y − z| : z ∈ M (11.2)
(11.2)
if and only if
(y− w,z) = 0 (11.3)
(11.3)
for all z ∈ M. Furthermore,
m∑
w = (y,xi)xi (11.4)
i=1
(11.4)
is the unique element of M which has this property. It is called the orthogonal projection.
Proof: First we show that if 11.3, then 11.2. Let z ∈ M be arbitrary. Then
Then a graph of this along with the graph of y = x^{2} is given below. In this graph, the dashed
graph is of y = x^{2} and the solid line is the graph of the above Fourier series approximation.
PICT
If we had taken the partial sum up to n much bigger, it would have been very hard to distinguish
between the graph of the partial sum of the Fourier series and the graph of the function it is
approximating. This is in contrast to approximation by Taylor series in which you only
get approximation at a point of a function and its derivatives. These are very close
near the point of interest but typically fail to approximate the function on the entire
interval.