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Chemical Engineering Hand Written_Notes_Part_18

# Chemical Engineering Hand Written_Notes_Part_18 - 38 2...

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38 2. FUNDAMENTALS OF FUNCTIONAL ANALYSIS is probably the most important result the plane geometry, is true in any inner product space. Lemma 2 . If x y in an inner product space then k x + y k 2 2 = k x k 2 2 + k y k 2 2 . Proof: k x + y k 2 2 = h x + y , x + y i = k x k 2 2 + k y k 2 2 + h x , y i + h y , x i . Definition 17 . (Orthogonal Set): A set of vectors S in an inner product space X is said to be an orthogonal set if x y for each x , y S and x 6 = y . 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 direc- tions in R n (b) sin ( nt ) and cos ( nt ) functions in C [0 , 2 π ] . 4.1. Gram-Schmidt procedure.

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Chemical Engineering Hand Written_Notes_Part_18 - 38 2...

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