Math 2406: Abstract Vector Spaces
Lecture 7 Plan
(Tuesday, September 11
Apostol 3.11-3.12.
Today we add an important ingredient to a linear space: an inner product, the generalization of a dot product for vectors. The inner product allows us to generalize our concept of length and angle, so we can talk about the "norm" of a function, and two functions being "orthogonal". Since the inner product allows us to measure things in a linear space, we call the pair (linear space, inner product) a metric space.Review questions:
1. Would the inner product in
example 2, p. 104, continue to be an inner product if the second term
were x_1*x_2?
2. In example 4, p. 105, what goes wrong if the weight function is w(t)
= sin(t)?
3. Is it true that if a function f
is orthogonal to a constant function on any fixed interval, then f has zero mean on that interval?
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