Chemical Engineering Hand Written_Notes_Part_16

# Chemical Engineering Hand Written_Notes_Part_16 - 34 2...

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34 2. FUNDAMENTALS OF FUNCTIONAL ANALYSIS Figure 1. Sequence of continuous functions vectors in R 3 ,say b x and b y , the angle between these two vectors is de f ned using inner (or dot) product of two vectors as cos( θ )=( b x ) T b y = μ x k x k 2 T y k y k 2 (4.1) = b x 1 b y 1 + b x 2 b y 2 + b x 3 b y 3 (4.2) Thefactthatcos ineo fang lebetweenanytwoun itvectorsisa lwayslessthan one can be stated as (4.3) | cos( θ ) | = |h b x , b y i| 1 Moreover, vectors x and y are called orthogonal if ( x ) T y =0 . Orthogonality is probably the most useful concept while working in three dimensional Euclid- ean space. Inner product spaces and Hilbert spaces generalize these simple geometrical concepts in three dimensional Euclidean space to higher or in f nite dimensional spaces. Definition 13 . (Inner Product Space): An inner product space is a linear vector space X together with an inner product de f ned on X × X .Co r r

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## This note was uploaded on 11/26/2011 for the course EGN 3840 taught by Professor Mr.shaw during the Fall '11 term at FSU.

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Chemical Engineering Hand Written_Notes_Part_16 - 34 2...

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