Question

Let D be a metric on X determined all constant k s.t
(k+d) is a metric. Give its Hint or prove it shortly.

Answer 1

Answer on Question #72876 – Math – Functional Analysis

**Question**

Let $d$ be a metric on a set $X$. Determine all constant $k$ such that $(k + d)$ is a metric on $X$. Give a hint or prove it shortly.

**Solution**

By the definition of a metric, the second condition should be satisfied:

$$x = y \Leftrightarrow (k + d)(x, y) = 0.$$

On the other hand, if $x = y$, then

$$(k + d)(x, y) = k + d(x, y) = k + 0 = k.$$

Therefore, $(k + d)$ is a metric only if $k = 0$. The converse proposition is also true.

**Answer:**

$$k = 0.$$

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