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Question: show that the la b inner product satisfies the following...

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Show that the L[a, b] inner product satisfies the following properties. . The L2 inner product is conjugate-symmetric (ie., 〈f, g , f)), homo- geneous, and bilinear (these properties are listed in Definition 0.1). Show that the L2 inner product satisfies positivity on the space of contin- uous functions on [a, b] by using the following outline. (a) We want to show that if bifa)j2 dt -0, then f() O for all a s t s b. (b) Suppose, by contradiction, that If (to)I > 0; then use the definition of continuity to show that If(t)| > If(to1/2 on an interval of the form [10-8, to + δ]. (c) Then show 4 which contradicts the assumption that J: If(t)12 dt = 0.

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