Normsノルム and the triangle inequality三角不等式さんかくふとうしき

1Introduction

The important point of this lecture is that a normノルム is a function that abstracts length, and the triangle inequality三角不等式さんかくふとうしき guarantees that it behaves like distance.

If we define $\parallel v\parallel =\sqrt{⟨v,v⟩}$\|v\|=\sqrt{\langle v,v\rangle} from an inner product内積ないせき, we still need to check that it really behaves like length: it should be nonnegative, scale by $\left|a\right|$|a| under scalar multiplication, and satisfy the rule that the length of a sum is no greater than the sum of the lengths. The last condition is the triangle inequality.

2Terms and definitions

A normノルム on a vector space $V$V is a function $\parallel ·\parallel$\|\cdot\| from $V$V to the nonnegative real numbers such that, for all $u,v\in V$u,v\in V and scalars $a$a,

and

The third inequality is the triangle inequality三角不等式さんかくふとうしき. A metric距離きょり can be defined from a norm by

3Plan

First we verify the norm axioms. Then we show that the norm induced by an inner product satisfies them. The triangle inequality is the only nontrivial part, and it follows from the Cauchy-Schwarz inequality.

4Intuitive explanation

The triangle inequality is the abstract form of the geometric fact that a detour is not shorter than the direct path. Moving by $u$u and then by $v$v is a path whose total length is $\parallel u\parallel +\parallel v\parallel$\|u\|+\|v\|, while moving directly to $u+v$u+v has length $\parallel u+v\parallel$\|u+v\|. Therefore

5Precise explanation

5.11. Defining a norm from an inner product

In an inner product space内積空間ないせきくうかん,

Positive definiteness gives $⟨v,v⟩\text{[PARSE ERROR: Undefined("Command("ge")")]}0$\langle v,v\rangle\ge0 and $⟨v,v⟩=0⟺v=0$\langle v,v\rangle=0\iff v=0, so nonnegativity and the zero-vector condition hold.

For a scalar $a$a,

so

5.22. Deriving the triangle inequality

In a real inner product space,

By Cauchy-Schwarz,

Therefore

Both sides are nonnegative, so taking square roots gives

In a complex inner product space, the expansion contains $2\mathrm{Re}⟨u,v⟩$2\operatorname{Re}\langle u,v\rangle, and the proof is the same because

6Concrete example

In $R^2$\mathbb R^2 with the standard inner product,

For $u=(1,0)$u=(1,0) and $v=(0,1)$v=(0,1),

so the triangle inequality reads $\sqrt{2}\text{[PARSE ERROR: Undefined("Command("le")")]}2$\sqrt2\le2.

7Another viewpoint

Once a norm is given,

defines distance. This supports notions such as convergence and error:

Geometrically, a norm is distance from the origin, and the triangle inequality ensures that this distance is compatible with geometric intuition.

8Criteria

• When length, distance, convergence, or error is involved, identify the norm.
• If an inner product is given, first construct $\parallel v\parallel =\sqrt{⟨v,v⟩}$\|v\|=\sqrt{\langle v,v\rangle}.
• For inequalities involving norms induced by inner products, Cauchy-Schwarz is the standard tool leading to the triangle inequality.

9Scope and limitations

Every norm obtained from an inner product satisfies the triangle inequality. However, not every norm comes from an inner product. For example, the ${\ell }^1$\ell^1 and ${\ell }^{\mathrm{\infty }}$\ell^\infty norms on $R^n$\mathbb R^n are important norms, but they are not generally the norm induced by the standard inner product.

10Final forms

11In one sentence

A normノルム abstracts length, and the triangle inequality三角不等式さんかくふとうしき makes that length behave as a distance.

12Exercise link

13Related links