Page 278 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 25: Orthonormal Sets

Notes
1 1
n 2 n 2
2 2
(x, e ) (y, e ) (By Cauchy inequality)
i i
i 1 i 1
x y 2 (by Bessel’s inequality for finite case)
2
n
(x, e ) (y, e ) x y … (1)
i
i
i 1
Finally let S is countably infinite. Let the vectors in S be arranged in a definite order as
S = {e , e , …, e , …}.
1 2 n
Let us define

(x, e ) (y, e ) = (x, e ) (y, e )
n .
i i n
i 1
But this sum will be well defined only if we can show that the series (x, e ) (y, e ) is
n
n
n 1
convergent and its sum does not change by rearranging its term i.e. by any arrangement of the
vectors in the set S.
Since (1) is true for every positive integer n, therefore it must be true in the limit. So

(x, e ) (y, e ) x y … (2)
n
n
n 1

From (2), we see that the series (x, e ) (y, e )
n is convergent. Since all the terms of the series
n
n 1
are positive, therefore it is absolutely convergent and so its sum will not change by any
rearrangement of its terms. So, we are justified in defining

(x, e ) (y, e ) = (x, e ) (y, e )
i i n n
n 1
and from (2), we see that this sum is x y .

25.2 Summary

 Two vectors in an inner product space are orthonormal if they are orthogonal and both of
unit length. A set of vectors from an orthonormal set if all vectors in the set are mutually
orthogonal and all of unit length.
 Examples of orthonormal sets are as follows:

n
(i) In the Hilbert space  , the subset e , e , …, e where e is the i-tuple with 1 in the i th
2
i
n
1
2
place and O’s elsewhere is an orthonormal set.
n
n
For (e , e) = 0 i j and (e , e) = 1 in the inner product x y of  .
i j i j i i 2
i 1
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