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Let B(x) be the set of all bounded functions defined on X defined by d(f,g)=sup x belong to X |f(x)-g(x)|, for all f, g that belong to X. Show that d is a metric

To show that d is a metric, we need to show that it satisfies the following properties:

1. Non-negativity: d(f,g) >= 0 for all f,g in X.
2. Identity of indiscernibles: d(f,g) = 0 if and only if f = g for all f,g in X.
3. Symmetry: d(f,g) = d(g,f) for all f,g in X.
4. Triangle inequality: d(f,g) + d(g,h) >= d(f,h) for all f,g,h in X.

Let's go through each property to show that d satisfies them:

1. Non-negativity:
Since the supremum of absolute values is always non-negative, it follows that d(f,g) = sup|x in X|f(x) - g(x)| >= 0.

2. Identity of indiscernibles:
If d(f,g) = 0, then sup|x in X|f(x) - g(x)| = 0 which means that for every ε > 0, there exists an x in X such that |f(x) - g(x)| < ε. This implies that f(x) = g(x) for all x in X, so f = g.

Conversely, if f = g, then for every x in X, |f(x) - g(x)| = 0. Therefore, sup|x in X|f(x) - g(x)| = 0, so d(f,g) = 0.

3. Symmetry:
For any f,g in X, d(f,g) = sup|x in X|f(x) - g(x)| = sup|x in X|g(x) - f(x)| = d(g,f).

4. Triangle inequality:
Let f,g,h be in X. We want to show that d(f,g) + d(g,h) >= d(f,h).

By the definition of suprema, for every ε > 0, there exists an x1 in X such that |f(x1) - g(x1)| < d(f,g) + ε, and there exists an x2 in X such that |g(x2) - h(x2)| < d(g,h) + ε.

Now consider the expression |f(x1) - g(x1) + g(x2) - h(x2)|. We can split this into |f(x1) - g(x1) + g(x2) - h(x2) + f(x2) - f(x2)| and use triangle inequality for the last two terms:

|f(x1) - g(x1) + g(x2) - h(x2) + f(x2) - f(x2)| <= |f(x1) - f(x2)| + |f(x2) - g(x2)| + |g(x2) - h(x2)|

Since |f(x1) - f(x2)| <= sup|x in X|f(x) - g(x)| = d(f,g), |f(x2) - g(x2)| <= sup|x in X|f(x) - g(x)| = d(f,g), and |g(x2) - h(x2)| <= sup|x in X|g(x) - h(x)| = d(g,h), we have:

|f(x1) - g(x1) + g(x2) - h(x2) + f(x2) - f(x2)| <= d(f,g) + d(f,g) + d(g,h) = 2d(f,g) + d(g,h).

Therefore, d(f,g) + d(g,h) >= 2d(f,g) + d(g,h) = d(f,g) + d(g,h), as desired.

Since d satisfies all the properties of a metric, we can conclude that d is indeed a metric.

To show that d is a metric, we need to verify the following properties:

1. Non-negativity: d(f,g) ≥ 0 for all f, g ∈ X.
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, g ∈ X.
4. Triangle inequality: d(f,g) ≤ d(f,h) + d(h,g) for all f, g, h ∈ X.

Let's prove each property:

1. Non-negativity: For any f, g ∈ X, by the definition of supremum, |f(x) - g(x)| ≥ 0 for all x ∈ X. Therefore, sup|x∈X|f(x) - g(x)| ≥ 0, which implies that d(f,g) ≥ 0.

2. Identity of indiscernibles: Suppose that d(f,g) = sup|x∈X|f(x) - g(x)| = 0. This means that |f(x) - g(x)| = 0 for all x ∈ X. Since the absolute value of a real number is non-negative, we have |f(x) - g(x)| = 0 if and only if f(x) - g(x) = 0 for all x ∈ X. Therefore, f(x) = g(x) for all x ∈ X, which implies that f = g.

3. Symmetry: Let f, g ∈ X. Then, sup|x∈X|f(x) - g(x)| is the same as sup|x∈X|g(x) - f(x)|, which is d(g,f). Therefore, d(f,g) = d(g,f) for all f, g ∈ X.

4. Triangle inequality: Let f, g, h ∈ X. By definition, d(f,g) = sup|x∈X|f(x) - g(x)| and d(f,h) = sup|x∈X|f(x) - h(x)|. Therefore, for any x ∈ X:

|f(x) - g(x)| ≤ |f(x) - h(x)| + |h(x) - g(x)|.

Taking the supremum on both sides of the inequality, we have:

sup|x∈X|f(x) - g(x)| ≤ sup|x∈X|f(x) - h(x)| + sup|x∈X|h(x) - g(x)|.

This implies that d(f,g) ≤ d(f,h) + d(h,g) for all f, g, h ∈ X.

Since all four properties hold true, we can conclude that d is a metric on the set of all bounded functions defined on X.

Did you know?

Did you know that in mathematics, there is a way to measure the similarity between different bounded functions? This is done using a concept called a metric. Let's consider a set X and define B(x) as the set of all bounded functions defined on X. We can then define a metric, denoted as d(f, g), which measures the difference or distance between two functions f and g. One interesting property of this metric is that it satisfies the conditions to be considered a true metric. To show this, we can use the concept of supremum, which is the least upper bound of a set. By taking the supremum of the absolute difference between f(x) and g(x) for all x in X, we obtain the metric d(f, g). This metric d(f, g) has several key properties. It is always a non-negative value, it equals zero if and only if f(x) is equal to g(x) for all x in X, it is symmetric (d(f, g) = d(g, f)), and it satisfies the triangle inequality (d(f, h) ≤ d(f, g) + d(g, h)). This important result showcases the existence of a metric that enables us to compare and measure the similarity between bounded functions. By understanding these mathematical concepts, we can gain deeper insights into the world of functions and their properties.