Show that f and g are orthogonal in the inner product space...

Show that $f$ and $g$ are orthogonal in the inner product space $C[a, b]$ with the inner product
$$\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x$$
$$C[-1,1], \quad f(x)=x, \quad g(x)=\frac{1}{2}\left(3 x^{2}-1\right)$$

Key Concepts

Even and Odd Functions in Integration

When integrating functions over symmetric intervals, the symmetry properties of even and odd functions are highly useful. An odd function integrated over an interval symmetric about zero yields zero, while an even function may yield a non-zero value. This property often simplifies the evaluation of integrals and is a key tool in verifying the orthogonality of functions defined on symmetric intervals.

Legendre Polynomials

Legendre polynomials form a sequence of orthogonal polynomials on the interval [-1, 1] with respect to the weight function 1. They arise in various contexts, particularly in problems involving spherical coordinates and potential theory. In this problem, the functions given correspond to the first few Legendre polynomials, demonstrating how orthogonality can be used to simplify computations in function spaces.

Orthogonality

Orthogonality in an inner product space means that two elements are perpendicular with respect to the inner product; that is, their inner product equals zero. For functions, this condition is expressed as the integral of their product over the designated interval being zero. This concept is fundamental in many areas of mathematics, especially in solving differential equations and in Fourier series.

Inner Product Space

An inner product space is a vector space equipped with an inner product—a function that takes two vectors (or functions) and returns a scalar. This metric allows us to define concepts such as length, angle, and orthogonality between elements. In the context of continuous functions on an interval [a, b], the inner product is often defined as the integral of the product of two functions over that interval.

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