Page 261 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes For, x y (x, y) = 0
(x,y) 0

(y, x) = 0
y x

(b) If x y then every scalar multiple of x is orthogonal to y i.e. x y x y for every scalar
C.
For, let be any scalar, then

( x, y) = (x, y)
= . 0
= 0
x y x y.
(c) The zero vector is orthogonal to every vector. For every vector x in H, we have

(0, x) = 0
0 x for all x H.
(d) The zero vector is the only vector which is orthogonal to itself. For,

2
if x x (x, x) = 0 x = 0 x = 0
Hence, if x x, then x must be a zero vector.

24.1.2 Pythagorean Theorem

Statement: If x and y are any two orthogonal vectors in a Hilbert space H, then
x + y = x – y = x + y .
2
2
2
2
Proof: Given x y (x, y) = 0, then we must have
y x i.e. (y, x) = 0
Now x + y 2 = (x + y, x + y)
= (x, x) + (x, y) + (y, x) + (y, y)
2
= x + 0 + 0 + y 2
2
= x + y 2
Also, x – y 2 = (x – y, x – y)
= (x, x) – (x, y) – (y, x) + (y, y)
= x – 0 – 0 – y 2
2
= x + y 2
2
2
2
x + y 2 = x – y = x + y 2
24.1.3 Orthogonal Sets

Definition: A vector x is to be orthogonal to a non-empty subset S of a Hilbert space H, denoted
by x S if x y for every y in S.

254 LOVELY PROFESSIONAL UNIVERSITY