Let R be the set of real numbers and d: R x R R be defined as follows. d(x, y) = 3|x - y®I. Then (R, d) is O a pseudo-metric space but not a metric space. O not a metric space and not a pseudo-metric space. O a metric space but not a pseudo-metric space. Oa metric space and a pseudo-metric space.

How do I show (i)? Please give an explanation with at most detail. I don't understand if it is too concise.

D. Consider R³ with the metric i. Verify that d is really a metric. ii. State and prove the Bolzano-Weierstrass Theorem for R³ with this metric. Hint: Examine the proof for the complex numbers! d((x1, x2, x3), (V1, V2, y3) = |x1-y₁| + x2 - y2 + x3 - y3|.

2. Consider a metric in a D-dimensional space. Simplify the following expressions, i.e. the result may be a function of D, a simple tensor that you should identify, or even a vanishing one. a) g gup b) 9 9μ c) (VHV-g¹ V²)Vv d) g (AB-B₁A₂) e) guga (AB - 9μo 9upº Bº)