Question

When does equality hold in the triangle inequality?

Answer 1

my attempt :

|x+y|≤|x|+|y|

|x+y|≤|x|+|y|

this implies

(|x+y|

)

2

=(|x|+|y|

)

2

⇒(x+y

)

2

=(|x|+|y|

)

2

⇒

x

2

+2xy+

y

2

=|x

|

2

+2|x||y|+|y

|

2

(|x+y|)2=(|x|+|y|)2⇒(x+y)2=(|x|+|y|)2⇒x2+2xy+y2=|x|2+2|x||y|+|y|2

since

x

2

=|x

|

2

x2=|x|2

⇒2xy=2|x||y|⇒xy=|x||y|

⇒2xy=2|x||y|⇒xy=|x||y|

and

|x⋅y|=|x|⋅|y| iff x=y

|x⋅y|=|x|⋅|y| iff x=y

therefore

xy=|xy|⇒xx=|xx|⇒

x

2

=|x

|

2

⇒x=x

xy=|xy|⇒xx=|xx|⇒x2=|x|2⇒x=x