Prove that ⟨f, g⟩=∫0^1 f(x) g(x) d x does not define an...

Prove that $\langle f, g\rangle=\int_0^1 f(x) g(x) d x$ does not define an inner product on the vector space $\mathrm{C}^0[-1,1]$. Explain why this does not contradict the fact that it defines an inner product on the vector space $\mathrm{C}^0[0,1]$. Does it define an inner product on the subspace $\mathcal{P}^{(n)} \subset \mathrm{C}^0[-1,1]$ consisting of all polynomial functions?

Key Concepts

Inner Product Properties

An inner product on a vector space is a bilinear (or sesquilinear, in complex spaces), symmetric mapping that assigns a real number to each pair of vectors, and it must satisfy linearity, symmetry, and positive definiteness. Positive definiteness means that the inner product of any nonzero vector with itself is strictly positive. This concept is foundational in linear algebra and functional analysis for defining notions of length and angle in abstract spaces.

Positive Definiteness and Degeneracy

The condition that ?f, f? = 0 if and only if f = 0 is essential for an inner product, known as positive definiteness. However, on symmetric intervals such as [–1,1], certain nonzero functions (for example, odd continuous functions that are not identically zero) may yield an inner product of zero due to cancellation effects in the integral. This failure of positive definiteness means that the bilinear form is degenerate and thus does not qualify as an inner product in that context.

Domain of Integration and Its Impact

The choice of the domain over which the integral is taken can significantly affect whether a given bilinear form qualifies as an inner product. For instance, integrating over [0,1] avoids the cancellation issues that occur on [–1,1] because the domain is not symmetric about zero, ensuring that ?f, f? > 0 for all nonzero continuous functions on [0,1]. This highlights the role of domain restrictions in establishing the necessary properties of inner products.

Subspace Considerations in Function Spaces

When dealing with subspaces of a larger function space, such as the space of polynomials of degree at most n, one must check if the inner product preserves the positive definiteness property within that subspace. Even though the polynomial subspace is finite-dimensional, if it includes functions like odd polynomials that integrate to zero on symmetric intervals, the inner product fails to be positive definite. This illustrates that the degenerate behavior seen in the full space may persist in certain subspaces unless additional restrictions are imposed.

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