Verification of the Given Statement in Metric Space: d(x, y) Σ from n1 to p of x^n - y^n in R^p

The provided problem involves verifying the given statement within the context of a metric space. Specifically, we are asked to verify whether the function d(x, y) Σ from n1 to p of x^n - y^n defines a metric space for vectors in R^p.

Understanding the Problem Statement

We are given the following statement:

Let X R^p and, for x, y ∈ R^p, define d(x, y) Σ from n1 to p of x^n - y^n.

The goal is to verify if d(x, y) satisfies the properties of a metric space. A metric space involving a function d(x, y) must satisfy the following axioms:

d(x, y) ≥ 0 for all x, y in R^p (non-negativity). d(x, y) 0 if and only if x y (identity of indiscernibles). d(x, y) d(y, x) for all x, y in R^p (symmetry). d(x, y) ≤ d(x, z) d(z, y) for all x, y, z in R^p (triangle inequality).

The most challenging part of verifying d(x, y) is to demonstrate that the triangle inequality holds.

Proving the Axioms for the Given Metric

1. Non-Negativity

To prove that d(x, y) ≥ 0 for all x, y ∈ R^p, observe that the expression x_n - y_n for each n is in the real numbers R. Since the sum of non-negative numbers is non-negative, it follows that:

d(x, y) Σ from n1 to p of |x^n - y^n| ≥ 0

2. Identity of Indiscernibles

To prove that d(x, y) 0 if and only if x y, observe that:

d(x, y) 0 implies that x^n - y^n 0 for all n. This means that x^n y^n for each n, and since the components of x and y must be identical, we conclude that x y.

3. Symmetry

To prove symmetry, observe that:

d(x, y) Σ from n1 to p of |x^n - y^n| Σ from n1 to p of |y^n - x^n| d(y, x)

4. Triangle Inequality

The most complex part is proving the triangle inequality. We need to show:

d(x, z) ≤ d(x, y) d(y, z)

Considering each component n, the inequality holds in R because:

Summing these inequalities from n1 to p, we get:

This confirms that d(x, y) satisfies the triangle inequality.

Since all four axioms are satisfied, d(x, y) defines a metric space on R^p.

Conclusion

In conclusion, the given function d(x, y) Σ from n1 to p of x^n - y^n indeed defines a metric space on R^p. The verification involves checking the non-negativity, identity of indiscernibles, symmetry, and the triangle inequality axioms of a metric space. For a comprehensive understanding and further research, one can refer to the literature on metric spaces and normed vector spaces.

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