Espaciamiento inicial de la plantación clonal.

(d) The inner product satisfies the Cauchy-Schwarz inequality |hf, gi| ≤ kfk kgk .

This inequality has a nice geometrical interpretation for real inner product spaces. In that case _hf, g_i=_hg, f_i is the familiar inner product and

−1_≤ hf, gi

kf_kkg_k ≡cos(angle between f and g)≤1 .

The Cauchy-Schwarz inequality follows from the fact that for any complex λ 0≤ hλf+g, λf +gi = |λ|2_kfk2_+kgk2

+ λhf, gi+λhg, fi . Lettingλ =x_|hhf,g_f,gi_i| we obtain for all real x

0_≤x2_kf_k2+ 2x_|hf, g_i|+_kg_k2 . Consequently, the discriminant,

|hf, gi|2_{− kfk}2_kgk2_,

of this quadratic expression must be negative or zero, otherwise this expres- sion would be negative for some values of x. It follows that

|hf, g_{i| ≤ k}f_{k k}g_k.

(e) The inner product implies the triangle inequality

kf ±gk ≤ kfk+kgk . (2.2) This inequality readily follows from the properties of the inner product (Why?)

2.2

Normed Linear Spaces

There exist other structures which a vector space may have. A norm on the vector space _V is a linear functional, say p(f), with the following three properties:

1. Positive definiteness: p(f) > 0 for all nonzero vectors f in _V, and p(f) = 0⇔f =~0.

2. Linearity: p(αf) =|α|p(f) for all vectors f and for all complex num- bersα.

3. Triangle inequality: p(f+g) ≤ p(f) +p(g) for all vectors f and g in V.

Such a function is usually designated by p(f) = _kf_k, a norm of the vector f. The existence of such a norm gives rise to the following definition:

A linear space _V equipped with a norm p(f) = _kf_k is called a normed linear space.

Example 1: Every inner product of an inner product space determines the norm given by

kfk= (hf, fi)12 ,

which, as we have seen, satisfies the triangle inequality, kf +g_{k ≤ k}f_k+_kg_k.

Thusan inner product space is always a normed linear space with the inner product norm. However, a normed linear space is not necessarily an inner product space.

Lecture 11

Example 2: Consider the vector space of n×n matrices A= [aij]. Then

kA_k= max

i,j |aij|

is a norm on this vector space.

Example 3: Consider the vector space of all infinite sequences x= (x1, x2, . . . , xk, . . .)

of real numbers satisfying the convergence condition ∞

X

k=1

2.2. NORMED LINEAR SPACES 65

where p_≥1 is a real number. Let the norm be defined by

kx_k= ∞ X k=1 |xk|p !1 p .

One can show that (Minkowski’s inequality) ∞ X k=1 |xk+yk|p !1 p ≤ ∞ X k=1 |xk|p !1 p + ∞ X k=1 |yk|p !1 p ,

i.e., that the triangle inequality,

kx+y_{k ≤ k}x_k+_ky_k,

holds. Hence k · kis a norm for this vector space. The space ofp-summable

_∞

P 1 |

xk|p <∞

real sequences equipped with the above norm is called ℓp

and the norm is called the ℓp_-norm.

This ℓp_{-norm gives rise to geometrical objects with unusual properties.}

consider the following

Example 4: The surface of a unit sphere centered around the origin of a linear space with the ℓp_{-norm is the locus of points {(x}

1, x2,· · · } for which ∞ X k=1 |xk|p !1 p = 1 .

Consider the intersection of this sphere with the finite dimensional subspace Rn_{, which is spanned by {(x}

1, x2,· · · , xn}.

a) When p= 2, this intersection is the locus of points for which |x1|2+|x2|2+· · ·+|xn|2 = 1 (unit sphere in Rn with ℓ2-norm)

This is the familiar (n− 1)-dimensional unit sphere in n-dimensional Eu- clidean space whose distance function is the Pythagorean distance

d(x, y) =_|x1−y1|2+|x2−y2|2+· · ·+|xn−yn|2 1 2 _.

x y

|x| +|y| =12 2

Figure 2.1: Circle inR2 _{endowed with the Pythagorean distance function of}

ℓ2_.

b) Whenp= 1, this intersection is the locus of points for which |x1|+|x2|+· · ·+|xn|= 1 (unit sphere in Rn with ℓ1-norm)

This is the (n₋ 1)-dimensional unit sphere in n-dimensional vector space endowed with a different distance function, namely one which is the sum of the differences

d(x, y) =_|x1−y1|+|x2 −y2|+· · ·+|xn−yn|,

between the two locations inRn_{, instead of the sum of squares. This distance}

function is called theHamming distance, and one must use it, for example, when travelling in a city with a rectangular grid of streets. With such a distance function a circle in R2 _{is a square standing on one of its vertices.}

See Figure 2.2. A 2-sphere in R3 _{is a cube standing on one of its vertices,}

etc.

c) Whenp→ ∞, this intersection is the locus of points for which

lim p→∞ n X k=1 |xk|p !1 p = 1 =_⇒

Max_{|x1|,|x2|,· · · ,|xn|} = 1 (unit sphere in Rn with ℓ∞-norm).

Such a unit sphere Rn _{is based on the distance function}

d(x, y) = lim p→∞ n X k=1 |xk−yk|p !1 p = Max_{|x1−y1|,|x2−y2|,· · · ,|xn−yn|} .