Unformatted text preview: piecewise
smooth functions which are periodic of period 2π . We say that V is a vector space because elements of V can be added and multiplied by scalars, these
operations satisfying the same rules as those for addition of ordinary vectors
and multiplication of ordinary vectors by scalars. We deﬁne an “inner product”
between elements of V by means of the formula
f, g = 1
π π f (t)g (t)dt.
−π Thus for example, if f (t) = sin t and g (t) = 2 cos t, then
f, g = 1
π π π 2 sin t cos tdt =
−π −π sin(2t)dt = − cos(2t)|π π = 0.
− The remarkable fact is that this inner product has properties quite similar
to those of the standard dot product on R n :
• f , g = g , f , whenever f and g are elements of V .
• f + g, h = f , h + g , h .
• cf, g = c f , g , when c is a real constant.
• f , f ≥ 0, with equality holding only if f = 0 (at all points of continuity).
This suggests that we might use geometric terminology for elements of V
just as we did for vectors in R n . Thus, for example, we will say that an...
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This document was uploaded on 01/12/2014.
- Winter '14