The 2-norm on Rk

You all know Rk as a vector space. We write

x = (x1, . . . , xk) ∈ Rk

We have the 2-norm on Rk

||x||2 =

√

√

√

√

k∑

i=1

x2i .

and the corresponding metric

d(x, y) = ||x − y||2.

. – p.1/24

Other norms

For any p ≥ 1 we define the p-norm on Rk by

||x||p =

(

k∑

i=1

|xi|p

)

1

p

.

In particular we have the 1-norm

||x||1 =k∑

i=1

|xi|.

And there is also the max-norm

||x||∞ = max|x1|, . . . , |xk|.

. – p.2/24

Important results

All norms on finite dimensional vector spaces are equivalent, in particular,for any p, q ∈ [1,∞] there are c, C > 0 such that for all x ∈ R

k

c||x||p ≤ ||x||q ≤ C||x||p.

Thus all norms induce the same topology (same open sets, sameconvergence concept, same continuous functions) on R

k.

The space Rk is a complete metric space when equipped with any of the

equivalent norms. This is extremely important and useful: if we can showthat (xn)n∈N is a Cauchy sequence then we know it converges – even ifwe have no idea about what he limit x is except that it is precisely the limitof xn for n → ∞.

. – p.3/24

An abstract point of view

The space Rk is identified with the space L(1, . . . , k, P(1, . . . , k), τ)

where τ is the counting measure;

f : k 7→ f(k) ! f = (f(1), . . . , f(k)).

Observe that for p ∈ [1,∞)

||f ||p =

(

k∑

i=1

|f(i)|p

)

1

p

=

(∫

|f |pdτ

)1

p

and that

||f ||∞ = maxi∈1,...,k

|f(i)|.

. – p.4/24

Infinite dimensional vector spaces

Let C([a, b]) denote the continuous functions defined on [a, b].

This space is equipped with a the max-norm

||f ||∞ = supx∈[a,b]

|f(x)|

Convergence of fn to f (fn, f ∈ C([a, b])) in the max-norm is equivalent touniform convergence of fn(x) to f(x) for all x ∈ [a, b].

The space C([a, b]) is an infinite dimensional vector space and equippedwith the metric from the max-norm it is a complete metric space.

. – p.5/24

Infinite dimensional vector spaces

Using the Riemann integral we can define the 1-norm of f ∈ C([a, b]) by

||f ||1 =

∫ b

a

|f(x)|dx.

The space C([a, b]) is not a finite dimensional vector space, and || · ||1 isnot equivalent to || · ||∞.

Obviously

||f ||1 ≤

∫ b

a

||f ||∞dx = (b − a)||f ||∞,

but we can also find a sequence fn with ||fn||1 = 1 for all n but||fn||∞ = n.

The space C([a, b]) equipped with the metric from the 1-norm is not acomplete metric space. . – p.6/24

Space of integrable functions

Let (X , E, µ) be a measure space.

We already know that L(X , E, µ) is a vector space:

If f, g ∈ L(X , E, µ) then f + g ∈ L(X , E, µ).

If f ∈ L(X , E, µ) and c ∈ R then cf ∈ L(X , E, µ).

Defining

||f ||1 =

∫

|f |dµ

we also know that

||f +g||1 =

∫

|f +g|dµ ≤

∫

|f |+ |g|dµ =

∫

|f |dµ+

∫

|g|dµ = ||f ||1 + ||g||1

and

||cf ||1 =

∫

|cf |dµ = |c|

∫

|f |dµ = |c|||f ||1.. – p.7/24

The 1-norm

We have showed that || · ||1 is a norm – except that

||f ||1 = 0 ⇔ f = 0 µ-a.e.

This implies in general that for the “metric” d(f, g) = ||f − g||1 we have

d(f, g) = 0 ⇔ f = g µ-a.e.

Such a “metric” is called a pseudo metric.

The norm satisfying f = 0 ⇒ ||f ||1 = 0 but not necessarily “⇐” is called asemi-norm.

. – p.8/24

Pseudo metrics

One can live happily with pseudo metrics. We can define balls, open sets(the topology), convergence and continuity, but there are two problems.

1. If xn → x and d(x, y) = 0 then xn → y – even if y 6= x

(non-uniqueness of the limit).

2. We have to rebuild the entire theory for metric spaces in terms ofpseudo metric spaces to assure exactly which results transfer fromthe theory for metric spaces and which do not.

In our case we can live with problem 1 – after all it means that if fn → f inthe 1-norm then in fact fn → g for all g = f µ-a.e.

. – p.9/24

A different solution

Let (X , E, µ) be a measure space and consider L(X , E, µ) equipped withthe semi-norm || · ||1.

Define the equivalence relation that

f ∼ g ⇔ ||f − g||1 =⇔ f = g µ-a.e.

Interpretation: If f 6= g but f ∼ g then f and g may be different from a

strict, function point-of-view, but from the point of view of the measure the

functions are identical. We collect functions that are equivalent in equi-

valence classes, and regard any function in an equivalence class as an

equally valid representation of that class.. – p.10/24

The space of equivalence classes

The equivalence relation ∼ gives a partition of the vector space L(X , E, µ)

in terms of equivalence classes.

Define

L(X , E, µ) = L(X , E, µ)/ ∼

as the set of equivalence classes, and for f ∈ L(X , E, µ) define

[f ] = g ∈ L(X , E, µ) | f ∼ g

Then we can define

[f ] + [g] = [f + g]

c[f ] = [cf ]

||[f ]||1 = ||f ||1

which turn L(X , E, µ) into a normed vector space.. – p.11/24

The abstract result

Let V be a vector space with a semi-norm || · ||. Introduce the equivalencerelation

x ∼ y ⇔ ||x − y|| = 0,

and define

[x] = y ∈ V | x ∼ y.

Theorem: Defining V = V/ ∼ as the set of equivalence classes one canintroduce addition and scalar multiplication on V by

[x] + [y] = [x + y]

c[x] = [cx],

which turn V into a vector space. Moreover, one can define the norm onV by

||[x]|| = ||x||,

which turns V into a normed space. . – p.12/24

The p-norms

Let (X , E, µ) be a measure space.

Definition: The space of p-double-integrable functions, denotedLp(X , E, µ), consists of those f ∈ M(X , E) with

∫

|f |pdµ < ∞.

Define the p-(semi)-norm

||f ||p =

(∫

|f |pdµ

)1

p

.

for f ∈ Lp(X , E, µ).

. – p.13/24

The p-norms

Let (X , E, µ) be a measure space.

Theorem: The set Lp(X , E, µ) is a vector space.

Question: But is || · ||p for p > 1 a norm?

. – p.14/24

Stochastic variables

Let (Ω, F, P ) be a probability space.

Then for p ∈ N the real valued, stochastic variable X is in Lp(Ω, F, P ) ifand only if it has finite p’th moment and

||X ||pp = E|X |p

is the absolute p’th moment of X .

Corollary 16.35 reads that for stochastic variables

||X ||q ≤ ||X ||p

for q = 1, . . . , p − 1.

. – p.15/24

|| · ||p is a (semi)-norm

Let (X , E, µ) be a measure space.

Obviously

||cf ||p =

(∫

|cf |pdµ

)1

p

=

(

|c|p∫

|f |pdµ

)1

p

= |c|

(∫

|f |pdµ

)1

p

= |c|||f ||p,

and we have

||f ||p = 0 ⇔ f = 0 µ-a.s..

But does the triangle inequality

||f + g||p ≤ ||f ||p + ||g||p

hold?. – p.16/24

Young’s inequality

We call p, q > 1 for dual exponents if

1

p+

1

q= 1.

Lemma: If p, q > 1 are dual exponents then for u, v ≥ 0

uv ≤up

p+

vq

q

. – p.17/24

Hölder’s inequality

Let (X , E, µ) be a measure space.

Theorem: If p, q > 1 are dual exponents, if f ∈ Lp(X , E, µ) and ifg ∈ Lq(X , E, µ) then fg ∈ L1(X , E, µ) and

||fg||1 =

∫

|fg|dµ ≤

(∫

|f |pdµ

)1

p

(∫

|g|qdµ

)1

q

= ||f ||p||g||q.

For p = q = 2 the inequality reads

∣

∣

∣

∣

∫

fgdµ

∣

∣

∣

∣

≤ ||f ||2||g||2,

which is usually known as Cauchy-Schwarz’ inequality.

. – p.18/24

Minkowski’s inequality

Let (X , E, µ) be a measure space.

Theorem: If f, g ∈ Lp(X , E, µ) for p ≥ 1 the

||f + g||p ≤ ||f ||p + ||g||p.

Thus || · ||p is a semi-norm on Lp(X , E, µ). The normed vector space ofequivalence classes is denoted

Lp(X , E, µ) = Lp(X , E, µ)/ ∼

where

f ∼ g ⇔ ||f − g||p = 0 ⇔ f = g µ-a.s.

. – p.19/24

Finite measure

Let (X , E, µ) be a measure space.

If µ(X ) < ∞ then the constant function 1 is q-double integrable for allq ≥ 1.

Take 1 ≤ r ≤ s, p = s/r and q = s/(s − r) then if f ∈ Ls we have |f |r ∈ Lp

and by Hölder’s inequality

||f ||rr =

∫

|f |rdµ ≤ |||f |r||p||1||q =

(∫

|f |sdµ

)r

s

µ(X )s−r

s

for

||f ||r ≤ µ(X )s−r

sr ||f ||s

In particular,

Lr ⊂ Ls

when µ(X ) < ∞.

. – p.20/24

Sums

Consider (N, P(N), τ).

The usual convention is then to write lp instead of Lp, and ifx = (x1, x2, . . .) ∈ lp then

||x||p =

(

∞∑

i=1

|xi|p

)1

p

.

Hölder’s and Mikowski’s inequalities specialize to non-trivial inequalitiesfor infinite sums – and by taking xk+1 = xk+2 = . . . = 0 they specialize tonon-trivial inequalities for finite sums:

k∑

i=1

|xiyi| ≤

(

k∑

i=1

|xi|p

)

1

p(

k∑

i=1

|yi|q

)

1

q

for1

p+

1

q= 1

(

k∑

i=1

|xi + yi|p

)

1

p

≤

(

k∑

i=1

|xi|p

)

1

p

+

(

k∑

i=1

|yi|p

)

1

p

. – p.21/24

Completeness

If (M, d) is a metric space we say that a sequence (xn)n∈N is Cauchy if wefor all ǫ > 0 can find N such that

d(xn, xm) < ǫ for n, m ≥ N.

We say that (M, d) is complete if all Cauchy sequences converges.

With (X , E, µ) a measure space, Lp(X , B, µ) is complete if we for anysequence (fn)n∈N of p-double-integrable functions fulfilling that for allǫ > 0 there is a N such that

||[fn] − [fm]||p = ||[fn − fm]||p = ||fn − fm||p < ǫ for n, m ≥ N

have that there exists a p-double integrable f such that

||[fn] − [f ]||p = ||[fn − f ]||p = ||fn − f ||p → 0

for n → ∞. . – p.22/24

Absolute convergence

Let V be a vector space with norm || · ||.

Definition: We say that a series∑k

i=1 xi in V is absolutely convergent if∑∞

i=1 ||xi|| < ∞.

Theorem: The normed vector space V is complete if and only if anyabsolutely convergent series

∑k

i=1 xi in V is convergent.

. – p.23/24

Completeness of Lp

Let (X , E, µ) be a measure space.

Theorem: The normed vector space Lp(X , B, µ) for p ≥ 1 is complete inthe metric given by the p-norm, that is, Lp(X , B, µ) is a Banach space.

Alternatively, if (fn)n∈N is a Cauchy sequence in Lp(X , B, µ) then thereexists a function f ∈ Lp(X , B, µ) such that

||fn − f ||p → 0

for n → ∞.

. – p.24/24