= 2. Consider the space C[-1, 1] with inner product (f, g) = f*. f(x)g(x) dx. A function f in this inner product space i

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2 Consider The Space C 1 1 With Inner Product F G F F X G X Dx A Function F In This Inner Product Space I 1 (88.64 KiB) Viewed 22 times

= 2. Consider the space C[-1, 1] with inner product (f, g) = f*. f(x)g(x) dx. A function f in this inner product space is called an even function in C1-77, 7) if f(-x) = f(x) for - < x < 0. A function f in this inner product space is called an odd function in C[-7, 7] if f(-x) = -f(x) for - <3 <. (a) Show that every even function in C[-1, 1] is orthogonal to every odd function in C[-1, 1] with respect to this inner product. Hint: review even and odd functions and their integrals. (b) Show that cos(mx) is orthogonal to sin(nx) for all integers m and n, with respect to this inner product. (c) Show that cos(mx) is orthogonal to cos(nx) for all integers m En, with respect to this inner product.