## January 12, 2012

### Obviousness – norm implies metric

We have a norm, which satisfies positivity, linearity and the triangle inequality. We have a metric, which satisfies positivity, symmetricity and the triangle inequality. We wish to prove that every norm is a metric. "Obviously", the only property we need to prove is... the triangle inequality.

At the time, I couldn't see how linearity implied symmetry was obvious. Now, I think it's about as obvious as the triangle inequality was - there's a step to take, even though it's simple.

Linearity requires that $||cx||=|c|||x||\text{ for } c \in \mathbb{C}$. Symmetricity requires that d(x,y)=d(y,x).

To prove it, we let d(x,y)= ||x-y|| = |-1|||x-y|| = ||-x-(-y)|| = ||y-x|| = d(y,x).

### 3 comments by 1 or more people

1. #### Nick

I agree entirely: finding that step is no easier than deducing the triangle inequality from the norm.

On this topic, I think the requirement that d(x,y)>=0 for all x & y in a metric space is actually unnecessary: we can prove it from the other axioms as follows:
2d(x,y)=d(x,y)+d(x,y)=d(x,y)+d(y,x) as d is symmetric
and d(x,y)+d(y,x)>=d(x,x) by the triangle inequality
and d(x,x)=0 so 2d(x,y)>=0 and hence d(x,y)>=0

12 Jan 2012, 12:19

2. #### Christopher Midgley

My first problem sheet goes slightly further: by modifying the triangle inequality requirement to be d(x,z)<=d(x,y)+d(z,y), we can remove the requirement for symmetry as well (it can be derived).

I’ll suppose it’s listed as required because it’s easier to check a metric satisfies symmetry than to check it satisfies the modified triangle inequality – if it fails, you can stop there.

14 Jan 2012, 11:30

3. #### Nick

Oh, I see: you can replace y with x to get d(x,z)<=d(z,x) (since d(x,x)=0)
and then by symmetry d(z,x)<=d(x,z)
so d(x,z)=d(z,x).

Thanks for pointing that out – it’s neat!

15 Jan 2012, 10:56

You are not allowed to comment on this entry as it has restricted commenting permissions.

## January 2012

Mo Tu We Th Fr Sa Su
Dec |  Today  | Feb
1
2 3 4 5 6 7 8
9 10 11 12 13 14 15
16 17 18 19 20 21 22
23 24 25 26 27 28 29
30 31

## Galleries

• Nice proof! Does this mean you're going to specialize in analysis and differential equations next ye… by Nick on this entry
• Hi Chris, It was most interesting to read your various reflections – thank you for sharing them. I'm… by Ceri Marriott on this entry
• Feel free. Chris by Christopher Midgley on this entry
• Hi Chris This is an honest final entry for the WSPA. Im glad that you have found the WSPA journey wo… by Samena Rashid on this entry
• Knowing the maximum price you would be comfortable with paying for X is extremely useful for compani… by Nick on this entry

Not signed in