Page 270 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 25: Orthonormal Sets

Thus a non-empty subset of a Hilbert space H is said to be an orthonormal set if it consists of Notes
mutually orthogonal unit vectors.

Notes
1. An orthonormal set cannot contain zero vector because 0 = 0.
2. Every Hilbert space H which is not equal to zero space possesses an orthonormal
set.
x
Since 0 x H. Then x 0. Let us normalise x by taking e = , so that
x
x 1
e = x = 1.
x x
e is a unit vector and the set {e} containing only one vector is necessarily an
orthonormal set.

x
3. If {x } is a non-empty set of mutually orthogonal vectors in H, then {e } = i is an
i i
x
i
orthonormal set.
25.1.3 Examples of Orthonormal Sets

n
th
1. In the Hilbert space  , the subset e , e , …, e where e is the i-tuple with 1 in the i place
2 1 2 n i
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
th
2. In the Hilbert space  , the set {e , e , …, e , …} where e is a sequence with 1 in the n place
2
n
1
n
2
and O’s elsewhere is an orthonormal set.
25.1.4 Theorems on Orthonormal Sets
Theorem 1: Let {e , e , …, e } be a finite orthonormal set in a Hilbert space H. If x is any vector in
1 2 n
H, then
n
2
2
(x, e ) x ; … (1)
i
i 1
further,
n
x (x, e ) e e for each j … (2)
i i j
i 1
Proof: Consider the vector
n
y = x (x, e ) e i
i
i 1

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