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f of V is of unit length if f , f = 1 and that two elements f and g of V are
perpendicular if f , g = 0.
In this terminology, the formulae (3.4), (3.5), and (3.6) can be expressed by
stating that the functions
1/ 2, cos t, cos 2t, cos 3t, . . . , sin t, sin 2t, sin 3t, . . .
are of unit length and perpendicular to each other. Moreover, by the theorem in
the preceding section, any element of of V can be written as a (possibly inﬁnite)
superposition of these functions. We will say that the functions
e0 (t) = √ ,
2 e1 (t) = cos t, e1 (t) = sin t,
ˆ e2 (t) = cos 2t, e2 (t) = sin 2t,
ˆ ... , ... make up an orthonormal basis for V .
We saw in Section 2.3 that if b1 , b2 , . . . , bn is an orthonormal basis for R n ,
f ∈ R n ⇒ f = (f · b1 )b1 + · · · + (f · bn )bn .
69 The same formula holds when R n is replaced by V and the dot product is
replaced by the inner product ·, · : If f is any element of V , we can write
f (t) = f (t), e0 (t) e0 (t) + f (t), e1 (t) e1 (t) + f (t), e2 (t) e2 (t) + · · ·
+ f (t), e1 (t)...
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This document was uploaded on 01/12/2014.
- Winter '14