Module 2 : Fundamentals of Vector Spaces
Section 4 : Inner Product Spaces and Hilbert Spaces
 

Definition 42 (Angle)
The angle $\theta$ between any two vectors in an inner product space is defined by MATH----------- (54)

Definition 43 (Orthogonal Vectors):
In a inner product space $X$ two vector MATH $\in X\,$are said to be orthogonal if MATHWe symbolize this by MATHA vector $\QTR{bf}{x}$ is said to be orthogonal to a set $S\,$ (written as $\QTR{bf}{x}\bot S$) if MATH for each $\QTR{bf}{z}$ $\in S.$

Just as orthogonality has many consequences in three dimensional geometry, it has many implications in any inner-product / Hilbert space Luen. The Pythagoras theorem, which is probably the most important result the plane geometry, is true in any inner product space.

Lemma 44
If MATH in an inner product space then MATH .

Proof: MATH

Definition 45 (Orthogonal Set):
A set of vectors $S$ in an inner product space $X$ is said to be an orthogonal set if MATH for each MATH $\in S$ and MATH The set is said to be orthonormal if, in addition each vector in the set has norm equal to unity.

Note that an orthogonal set of nonzero vectors is linearly independent set. We often prefer to work with an orthonormal basis as any vector can be uniquely represented in terms of components along the orthonormal directions. Common examples of such orthonormal basis are (a) unit vectors along coordinate directions in $R^{n}$ (b) function MATH and MATH in $L_{2}[0,2\pi].$