Question

Let X = C[0, 1] . Define d : X \times X \rightarrow R by d(f,g) = \int_{0}^{1}\mid f(t) -g(t) \mid dt , f, g \in X where the integral is the Riemann integral. Show that d is a metric on X . Find d(f,g) wheref(x) = 4x and g(x) = x^3 , x \in [0,1]

26 Feb 2024
Answer :
Word Count : 420

To show that \( d \) is a metric on \( X \), we need to verify the three properties of a metric:

1. **Non-negativity**: \( d(f, g) \geq 0 \) for all \( f, g \in X \), with equality if and only if \( f = g \).
2. **Identity of indiscernibles**: \( d(f, g) = 0 \) if and only if \( f = g \).
3. **Symmetry**: \( d(f, g) = d(g, f) \) for all \( f, _____ _________ _______ _______ ___ _________ ________ _______ ___.
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Let X = C[0, 1] . Define d : X \times X \rightarrow R by d(f,g) = \int_{0}^{1}\mid f(t) -g(t) \mid dt , f, g \in X where the integral is the Riemann integral. Show that d is a metric on X . Find d(f,g) wheref(x) = 4x and g(x) = x^3 , x \in [0,1]

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