24.3 Approximations In Hilbert Space
The Gram Schmidt process applies in any Hilbert space.
Theorem 24.3.1 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 so
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,
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 vectors. Suppose
and yk → y ∈ H.
y ∈ M
Then let ck ≡
is a Cauchy sequence in
and so it converges to c ∈ Fn.
Theorem 24.3.2 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 24.1.13 this verifies 24.13.
The square of the distance, d is given by
and this shows 24.14
class=”left” align=”middle”(U)24.4. ORTHONORMAL SETS