Plan 2 Outline: Basic definitions and facts Relation between L p , - - PowerPoint PPT Presentation

Function spaces Lp

Updated April 22, 2020

Plan

2

Outline: Basic definitions and facts Relation between Lp, interpolation of norms Uniform convexity of Lp with 1 ă p ă 8 Projections on closed convex sets Peculiarities of Lp-spaces with 0 ď p ă 1 L2-space as inner product space

Definition

3

Definition Given measure space pX, F, µq, for p P p0, 8q let LppX, F, µq :“ ! f : X Ñ R: F-measurable ^ ż |f|pdµ ă 8 ) . For p :“ 0 we set L0pX F, µq :“

• f : X Ñ R: F-measurable

( and for p :“ 8 we let L8pX F, µq :“ ! f P L0 : ` DM P r0, 8q: µp|f| ą Mq “ 0q ˘) Alternative notations: Lp, Lppµq or LppXq When X is countable and µ is the counting measure: ℓppXq

Vector space

4

Lemma For each p P r0, 8s and each a, b P R, f, g P Lp ñ af ` bg P Lp. In particular, Lp is a vector space over R. Integrability condition: use |f ` g|p ď 2p|f|p ` 2p|g|p

Lp-norm

5

Lemma For f P L0 and for p P r1, 8q let }f}p :“ ´ ż |f|pdµ ¯1{p and for p :“ 8 set }f}8 :“ inf

• M P r0, 8q: µp|f| ą Mq “ 0

( Then } ¨ }p is a seminorm on Lp such that }f}p “ 0 ñ f “ 0 µ-a.e. In particular, } ¨ } is the norm on trfs: f P Lpu where rfs :“ tg P Lp : g “ f µ-a.e.u and }rfs}p :“ }f}p.

Proof

6

Homogeneity checked directly. For triangle inequality, we proceed by showing: Young’s inequality: @α P r0, 1s @x, y P r0, 8q: xαy1´α ď αx ` p1 ´ αqy H¨

• lder’s inequality: @f, g P L0 @p, q P r1, 8s:

1 p ` 1 q “ 1 ñ ż |fg|dµ ď }f}p}g}q. Minkowski’s inequality: @p P r1, 8q @f, g P Lp : } f ` g}p ď }f}p ` }g}p The case of p “ 8 is checked directly or via limits using @f P L0 : lim

pÑ8 }f}p “ }f}8

Banach space structure

7

Theorem (Completeness of Lp-spaces) For each p P r1, 8s and any tfnuně1 Ď Lp that is Cauchy in the sense lim

NÑ8 sup m,něN

}fn ´ fm}p “ 0 there exists f P Lp such that lim

nÑ8 }fn ´ f}p “ 0.

Moreover, we then have }f}p “ lim

nÑ8 }fn}p

where the limit on the right exists.

Proof

8

Assume p ă 8, pick tnkukě1 with supm,něnk }fn ´ fm}p ď 2´k. Define Fn :“

n

ÿ

k“1

|fnk`1 ´ fnk| ^ F :“ ÿ

kě1

|fnk`1 ´ fnk| Then }Fn}p ď 1 and, by Fatou’s lemma, }F}p ď 1. So F ă 8 µ-a.e. and so f :“ fn1 ` ÿ

kě1

` fnk`1 ´ fnk ˘ absolutely convergent µ-a.e. Telescoping: fnk Ñ f µ-a.e. and so, by Fatou again, }fnk}p Ñ }f}p and }f ´ fnk}p ď lim sup

jÑ8

}fnj ´ fnk}p ď 2´k This gives fn Ñ f in Lp. The case p “ 8 checked directly.

Two corollaries

9

Corollary If fn Ñ f in Lppµq then there exists a subsequence tnkukě1 such that sup

kě1

|fnk| P Lppµq and fnk Ý Ñ

kÑ8 f µ-a.e.

Corollary For each p P r1, 8s, the space trfs: f P Lpu endowed with norm } ¨ }p is a complete normed vector space a.k.a. Banach space.

Relations between Lp-spaces

10

Lemma Let Lp “ LppX, F, µq. (1) If µ is finite then p ÞÑ Lp is non-increasing, @p, r P r0, 8s: p ă r ñ Lr Ď Lp (2) If X is countable and µ is the counting measure, then p ÞÑ ℓppXq is non-decreasing, @p, r P r0, 8s: p ă r ñ ℓppXq Ď ℓrpXq Neither of these hold in general measure spaces but we have @p, q, r P p0, 8s: q ă p ă r ñ Lp Ď Lq ` Lr where Lq ` Lr :“ tf ` g: f P Lq ^ g P Lru.

Proof

11

If µ finite, @p ą q ą 0: }f}q ď µpXq1{q´1{p }f}p so p ÞÑ Lp is non-increasing. For X countable and µ the counting measure, @r ą p ą 0: }f}r ď }f}1´p{r

8

}f}p{r

p

so p ÞÑ ℓppXq is non-decreasing. For general case: f “ f1t|f|ě1u ` f1t|f|ď1u and use above for the parts.

Interpolation of p-norms

12

Lemma For any q, p, r P p0, 8s with q ă p ă r we have }f}p ď }f}α

q}f}1´α r

where α P p0, 1q is the unique number such that 1 p “ α q ` 1 ´ α r . In particular, Lq X Lr Ď Lp. Proof: ż |f|pdµ “ ż |f|αp |f|p1´αqpdµ ď }f αp} q

αp }f p1´αqp} r p1´αqp

Uniform convexity

13

Definition A normed linear space V is uniformly convex if for each ǫ ą 0 there is δ ą 0 such that for all x, y P V with }x} “ 1 “ }y}, }x ´ y} ě ǫ ñ › › ›x ` y 2 › › › ď 1 ´ δ Says that unit ball is strictly convex and “uniformly curved”. Fails in L1 and L8, but: Theorem (Clarkson 1936) For each p P p1, 8q, the normed linear space Lp is uniformly convex. Proof based on Clarkson’s inequalities. We will follow the proof by O. Hanner (1956)

Hanner’s inequalities

14

Lemma For all p P r1, 2s and all f, g P Lp, we have }f ` g}p

p ` }f ´ g}p p ě

` }f}p ` }g}p ˘p ` ˇ ˇ}f}p ´ }g}p ˇ ˇp and 2p` }f}p

p ` }g}p p

˘ ě ` }f ` g}p ` }f ´ g}p ˘p ` ˇ ˇ}f ` g}p ´ }f ´ g}p ˇ ˇp The inequalities are reversed for p P r2, 8q. Note: 2nd inequality obtained from 1st by subs f Ñ f ` g ^ g Ñ f ´ g Generalize parallogram law for p “ 2

Proof: main steps

15

Suppose p P r1, 2s (for p P r2, 8q all inequalities reversed). Key identity: @t P r0, 1s @u, v P R: |u ` v|p ` |u ´ v|p ě aptq|u|p ` bptq|v|p where aptq :“ p1 ` tqp´1 ` p1 ´ tqp´1 and bptq :“ “ p1 ` tqp´1 ´ p1 ´ tqp´1‰ t1´p Proved by calculus. Apply to u “ fpxq and v “ gpxq and integrate: }f ` g}p

p ` }f ´ g}p p ě aptq}f}p p ` bptq}g}p p

As aptq ` bptqtp “ p1 ` tqp ` p1 ´ tqp, setting t :“ }g}p{}f}p we then get the result.

Proof of uniform convexity

16

Suppose p P r2, 8q. For }f}p “ }g}p “ 1, the 1st inequality gives › › ›f ` g 2 › › ›

p ď

ˆ 1 ´ }f ´ g}p

p

2p ˙1{p so claim holds with δ :“ 1 ´ p1 ´ pǫ{2qpq1{p For p P r1, 2s assume WLOG }f ` g}p ě }f ´ g}p. The 2nd inequality gives 2p`1 ě }f ` g}p

p h

´}f ´ g}p }f ` g}p ¯ where hpxq :“ p1 ` xqp ` p1 ´ xqp. Calculus shows hpxq ě 2 ` cppqx2 where cppq :“ ppp ´ 1q This gives › › ›f ` g 2 › › ›

p ď

´ 1 ` cppq 8 }f ´ g}2

p

¯´1 and claim follows with δ :“ 1 ´ p1 ´ 1

8cppqǫ2q´1.

Projections on closed convex sets

17

Proposition (Projection on closed convex subsets) Let p P p1, 8q and let C Ď Lp be non-empty and convex, i.e., @f, g P C @α P r0, 1s: αf ` p1 ´ αqg P C and closed (i.e., with all Lp-Cauchy sequences in C convergent in C). Let f P Lp C. Then C contains an element that is closest to f, Dh P C: }f ´ h}p “ inf

gPC }f ´ g}p

Proof of Proposition

18

By shift suppose f “ 0 and C Ď Lp closed convex with 0 R C. Let thnuně1 be s.t. lim

nÑ8 }hn}p “ inf gPC }g}p

Convexity of C ensures }hn}p ` }hm}p 2 ě › › ›hn ` hm 2 › › ›

pě inf gPC }g}p

So lim

NÑ8

sup

m,něN

› › ›hn ` hm 2 › › ›

p“ inf gPC }g}p

For normalized functions h1

n :“ hn{}hn}p this implies

lim

NÑ8 sup m,něN

› › ›h1

n ` h1 m

2 › › ›

p“ 1

Uniform convexity: th1

nuně1 Cauchy and thus convergent in Lp.

Then: thnuně1 convergent to some h P C, }h}p “ infgPC }g}p.

Lp-spaces with 0 ă p ă 1

19

Lemma (Reverse Minkowski inequality) For each p P p0, 1q, we have @f, g P Lp : › ›|f| ` |g| › ›

p ě }f}p ` }g}p

The inequality is strict whenever µpf ¨ g ‰ 0q ą 0. Proof: As x ÞÑ xp concave, so for a, b ě 0, t P p0, 1q: pa ` bqp “ ´ ta t ` p1 ´ tq b 1 ´ t ¯p ě t1´pap ` p1 ´ tq1´tbp True for t “ 0, 1 by continuity, strict if a, b ą 0 and t P p0, 1q. Now set a :“ |fpxq|, b :“ |gpxq| and integrate to get }|f| ` |g|}p

p ě t1´p}f}p p ` p1 ´ tq1´p}g}p

Finally, take t :“ }f}p{p}f}p ` }g}pq.

Lp (still) a complete metric space

20

Lemma For p P p0, 1q and f, g P Lppµq, define ̺ppf, gq :“ }f ´ g}p

p “

ż |f ´ g|pdµ Then ̺p is a pseudometric with ̺ppf, gq “ 0 ô f “ g µ-a.e. In particular, ̺p is a metric on the factor space trfs: f P Lpu and the resulting metric space is complete. Proof: Triangle inequality because @a, b ě 0: pa ` bqp ď ap ` bp Completness: same argument as for p ě 1.

Convexity curse

21

Ball in Lp, 0 ă p ă 1, not convex! Proposition Let p P p0, 1q and consider the Lp space over pX, F, µq such that for each δ ą 0 there are disjoint tAnuně1 Ď F such that X “ ď

ně1

An ^ sup

ně1

µpAnq ă δ Then every open, convex subset of pLp, ̺pq is either empty or Lp itself Convexity = algebra, open = topology so: Lp cannot be remetrized to a locally-convex vector space (let alone Banach space)

Proof

22

U Ď Lp non-empty, convex, 0 P U by shift. As U open, there is r ą 0 s.t. Bp0, rq Ď U. Let f P Lp non-zero. Set θ :“ 1{}f}p

• p. Integrability of |f|p +

assumption on measure space ñ DtAnuně1 partition s.t. @n ě 1: ż

An

|f|pdµ ă 1 θ prθq

1 1´p

Dominated Convergence: DN ě 1 ż

Ť

něN An

|f|p dµ ă 1 θ prθq

1 1´p

Set A1

n :“ An for n ă N and A1 N :“ Ť něN An and . . .

Proof continued

23

. . . let αn :“ θ ż

A1

n

|f|pdµ, and set hn :“ # α´1

n 1A1

n f,

if αn ‰ 0, 0, else. Then řN

n“1 αn “ 1 and αn ă prθq

1 1´p . Now

̺pp0, hnq “ 1 αp

n

ż

A1

n

|f|pdµ “ α1´p

n

θ ă rθ θ “ r and so hn P Bp0, rq Ď U. But U is convex and f “

N

ÿ

n“1

αnhn. so f P U as well. Hence U “ Lp.

No continuous linear functions on Lp!

24

Corollary Let p P p0, 1q. Under the condition on pX, F, µq in Proposition, LppX, F, µq admits no non-constant continuous convex (and, in particular, linear) functional. Proof: If φ: Lp Ñ R is a convex continuous map then tf P Lp : φpfq ă au is open and convex. Analysis without continuous linear function hard . . . On ℓppNq continuous linear functionals exist but they are (unique) extensions of those on ℓ1pNq, which is dense in ℓppNq.

Space L0

25

Endow with convergence in measure defined by µp|fn ´ f| ą ǫq Ñ 0 @ǫ ą 0. For µ finite, topology defined by neighborhoods:

• h P L0 : µp|h ´ f| ą ǫq ă ǫ

( indexed by f P L0 and ǫ ą 0. Metrized by Ky Fan metric: ̺0pf, gq :“ inf

• ǫ ě 0: µp|f ´ g| ą ǫq ď ǫ

( Lemma Suppose µ is finite. Then ̺0 is a pseudo-metric on L0 and is a metric

• n the factor space trfs: f P L0u. The factor space is complete.

Special role of L2

26

Norm } ¨ }2 arises from inner product xf, gy :“ ż fg dµ. via }f}2 “ xf, fy1{2. (Conjugate g if over C.) Paralellogram law (p “ 2 version of Hanner’s inequality): @f, g P L2 : }f ` g}2

2 ` }f ´ g}2 2 “ 2}f}2 2 ` 2}g}2 2

Special role of L2 continued ...

27

Inner product reconstructed from norm: Lemma (Polarization identity) In L2-spaces over R, for all f, g P L2 we have xf, gy “ 1 2}f ` g}2

2 ´ 1

2}f ´ g}2

2

In L2-spaces over C, we instead have xf, gy “ 1 4}f ` g}2

2 ´ 1

4}f ´ g}2

2 ` i

4}f ` ig}2

2 ´ i

4}f ´ ig}2

2

Proof: Plug in formula for norm using inner product

Conclusion

28

Paralellogram law necessary and sufficient for norm to arise from inner product: Lemma A norm } ¨ } arises from an inner product x¨, ¨y if and only if it satisfies the parallelogram law. Orthonormal bases, Fourier analysis (abstract and concrete), etc to be discussed in context of Hilbert spaces.