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(d) Let d be any metrie on Rand define p by plz, v) = Show that pis also a metrie on R.

(d) Let d be any metrie on Rand define p by plz, v) = Show that pis also a metrie on R.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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or (a) Investigate whether d: RxR -R detitned s d(z, p) = (r- yj* is a metrie on tlhe set al all real uumbers? (b) Prove the generalised triangle inequality below Id(z, u) – d(z, w)| S d(z, =) + diy, w) (e) Using the triangle inequality, show that Idf z. =) - d(y. 3)|S d(r, y) (d) Let d be any metrie on R and define p by plz, w) 3= s.y) Show that pis also a etrie on R. (e) Pruve that tlhe funetion d:R" x R" -R* delined by d(x, ) = V(E - ) + (- +- +(- W) isa tmetrie on R", where z= (, 2. n) and y = (- a, ... ) are paints in R".
Transcribed Image Text:or (a) Investigate whether d: RxR -R detitned s d(z, p) = (r- yj* is a metrie on tlhe set al all real uumbers? (b) Prove the generalised triangle inequality below Id(z, u) – d(z, w)| S d(z, =) + diy, w) (e) Using the triangle inequality, show that Idf z. =) - d(y. 3)|S d(r, y) (d) Let d be any metrie on R and define p by plz, w) 3= s.y) Show that pis also a etrie on R. (e) Pruve that tlhe funetion d:R" x R" -R* delined by d(x, ) = V(E - ) + (- +- +(- W) isa tmetrie on R", where z= (, 2. n) and y = (- a, ... ) are paints in R".
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