17.3 Approximations In Hilbert Space
The Gram Schmidt process applies in any Hilbert space.
Theorem 17.3.1 Let
be a basis for M a subspace of H a Hilbert space.
Then there exists an orthonormal basis for M,
which has the property that
for each k ≤ n, span
. Also if
is a closed subspace.
be a basis for
Let u1 ≡ x1∕
Thus for k
is an orthonormal set. Now suppose for some
k < n, u1
have been chosen such that
where the denominator is not equal to zero because the xj form a basis and
Thus by induction,
Also, xk+1 ∈ span
which is seen easily by solving
If l ≤ k,
generated in this way are therefore an orthonormal basis because
each vector has unit length.
Consider the second claim about finite dimensional subspaces. Without loss
of generality, assume
is linearly independent. If it is not, delete
vectors until a linearly independent set is obtained. Then by the first part,
where the ui
are an orthonormal set of
and yk → y ∈ H.
Is y ∈ M
Then let ck ≡
is a Cauchy sequence in
and so it converges to c ∈ Fn.
This completes the proof.
Theorem 17.3.2 Let M be the span of
in a Hilbert space, H and let
y ∈ H. Then Py is given by
and the distance is given by
It follows that
for all u ∈ M and so by Corollary 17.1.13 this verifies 17.3.17.
The square of the distance, d is given by
and this shows 17.3.18
What if the subspace is the span of vectors which are not orthonormal? There is a
very interesting formula for the distance between a point of a Hilbert space and a finite
dimensional subspace spanned by an arbitrary basis.
Definition 17.3.3 Let
⊆ H, a Hilbert space. Define
Thus the ijth entry of this matrix is
. This is sometimes called the Gram matrix.
Also define G
as the determinant of this matrix, also called the Gram
The theorem is the following.
Theorem 17.3.4 Let M = span
⊆ H, a Real Hilbert space where
is a basis and let y ∈ H. Then letting d be the distance from y to
Proof: By Theorem 17.3.1 M is a closed subspace of H. Let ∑
k=1nαkxk be the
element of M which is closest to y. Then by Corollary 17.1.13,
for each p = 1,2,
This yields the system of equations,
Also by Corollary 17.1.13,
and so, using 17.3.22,
Then 17.3.22 and 17.3.23 imply the following system
By Cramer’s rule,
and this proves the theorem.
+ class=”left” align=”middle”(U)17.4. THE MÜNTZ THEOREM