Question

Show that the L²[a, b] inner product satisfies the following properties. • The L² inner product is conjugate-symmetric (i.e., ?f, g? = ??g, f?), homogeneous, and bilinear (these properties are listed in Definition 0.1). • Show that the L² inner product satisfies positivity on the space of continuous functions on [a, b] by using the following outline. (a) We want to show that if ??? |f(t)|² dt = 0, then f(t) = 0 for all a ? t ? b. (b) Suppose, by contradiction, that |f(t?)| > 0; then use the definition of continuity to show that |f(t)| > |f(t?)|/2 on an interval of the form [t? - ?, t? + ?]. (c) Then show ??? |f(t)|² dt ? (|f(t?)|² / 4)[2?] > 0, which contradicts the assumption that ??? |f(t)|² dt = 0.

          Show that the L²[a, b] inner product satisfies the following properties.
• The L² inner product is conjugate-symmetric (i.e., ?f, g? = ??g, f?), homogeneous, and bilinear (these properties are listed in Definition 0.1).
• Show that the L² inner product satisfies positivity on the space of continuous functions on [a, b] by using the following outline.
(a) We want to show that if ??? |f(t)|² dt = 0, then f(t) = 0 for all a ? t ? b.
(b) Suppose, by contradiction, that |f(t?)| > 0; then use the definition of continuity to show that |f(t)| > |f(t?)|/2 on an interval of the form [t? - ?, t? + ?].
(c) Then show
??? |f(t)|² dt ? (|f(t?)|² / 4)[2?] > 0,
which contradicts the assumption that ??? |f(t)|² dt = 0.

Show that the L²[a, b] inner product satisfies the following properties.
• The L² inner product is conjugate-symmetric (i.e., ?f, g? = ??g, f?), homogeneous, and bilinear (these properties are listed in Definition 0.1).
• Show that the L² inner product satisfies positivity on the space of continuous functions on [a, b] by using the following outline.
(a) We want to show that if ??? |f(t)|² dt = 0, then f(t) = 0 for all a ? t ? b.
(b) Suppose, by contradiction, that |f(t?)| > 0; then use the definition of continuity to show that |f(t)| > |f(t?)|/2 on an interval of the form [t? - ?, t? + ?].
(c) Then show
??? |f(t)|² dt ? (|f(t?)|² / 4)[2?] > 0,
which contradicts the assumption that ??? |f(t)|² dt = 0.

Added by Pedro D.

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