Advanced Functional Analysis
Locally Convex Spaces and Spectral Theory
Andreas Kriegl
This is the preliminary english version of the script for my homonymous lecture course in the Sommer Semester 2019. It was translated from the german original using a pre and post processor (written by myself) for google translate. Due to the limitations of google translate – see the following article by Douglas Hofstadter www.theatlantic.com/. . . /551570 – heavy corrections by hand had to be done af- terwards. However, it is still a rather rough translation which I will try to improve during the semester.
The contents of this lecture course are choosen according to the curriculum of the Master’s program: locally convex vector spaces as well as bounded and unbounded operators on Hilbert spaces. These two topics are only loosely related to each other and this dichotomy is reflected in these lecture notes.
The first part deals with an introduction to the theory of locally convex spaces. In addition to the basic concepts and constructions, we will discuss generalizations of the central propositions of Banach-space theory and discuss the duality theory.
The second part revolves around the spectral theory of bounded and unbounded operators. I followed closely the chapters VII - X in [5].
These lecture notes are the result of a combination of lecture notes for lectures I have given in the years since 1991.
Corrections to the predecessor versions I owe (in chronological order) to Andreas Cap, Wilhelm Temsch, Bernhard Reiscker, Gerhard Totschnig, Leonhard Summer- er, Michaela Mattes, Muriel Niederle, Martin Anderle, Bernhard Lamel, Konni Ri- etsch, Oliver Fasching, Simon Hochgerner, Robert Wechsberg, Harald Grobner, Johanna Michor, Katharina Neusser and David Wozabal. I would like to take this opportunity to thank them most sincerely.
Vienna, February 2012, Andreas Kriegl Franz Berger provided me with a comprehensive correction list in September 2012.
Sarah Koppensteiner provided me with another comprehensive correction list in November 2015.
Inhaltsverzeichnis
I Locally Convex Spaces 4
1. Seminorms 5
1.1 Basics 5
1.2 Important norms 6
1.3 Elementary properties of seminorms 9
1.4 Seminorms versus topology 12
1.5 Convergence and continuity 16
1.6 Normable spaces 18
2. Linear mappings and completeness 19
2.1 Continuous and bounded mappings 19
2.2 Completeness 22
3. Constructions 27
3.1 General initial structures 27
3.2 Products 30
3.3 General final structures 35
3.4 Finite dimensional lcs 38
3.5 Metrizable lcs 40
3.6 Coproducts 42
3.7 Strict inductive limits 46
3.8 Completion 47
3.9 Complexification 49
4. Baire property 54
4.1 Baire spaces 54
4.2 Uniform boundedness 59
4.3 Closed and open mappings 63
5. The Theorem of Hahn Banach 67
5.1 Extension theorems 67
5.2 Separation theorems 71
5.3 Dual spaces of important examples 72
5.4 Introduction to duality theory 77
5.5 Compact sets revisited 85
II Spectral Theory 90
6. Spectral and Representation Theory for Banach Algebras 91
Preliminary remarks 91
Recap from complex analysis 100
Functional Calculus 110
Dependency of the spectrum on the algebra 115
Commutative Banach algebras 116
7. Representation theory for C˚-algebras 123
Basics about C˚-algebras 123
Spectral Theory of Abelian C˚-Algebras 126
Applications to Hermitian elements 130
Ideals and quotients ofC˚-algebras 133
Cyclic representations ofC˚-algebras 138
Irreducible representations ofC˚-algebras 143
Group Representations 146
8. Spectral theory for normal operators 164
Representations of Abelian C˚-algebras and spectral measures 164
Spectral theory for normal operators 174
Spectral theory of compact operators 178
Normal operators as multiplication operators 181
Commutants and von Neumann algebras 187
Multiplicity Theory for Normal Operators 197
9. Spectral theory for unbounded operators 202
Unbounded Operators 202
Adjoint operator 203
Invertibility and spectrum 211
Symmetric and self adjoint operators 214
Spectrum of symmetric operators 217
Symmetrical extensions 219
Cayley Transformation 223
Unbounded normal operators 225
1-parameter groups and infinitesimal generators 234
Literaturverzeichnis 242
Index 244
Teil I
Locally Convex Spaces
1. Seminorms
In this chapter we will introduce the adequate notion of distance on vector spaces and discuss its elementary properties.
1.1 Basics
1.1.1 Motivation and definitions.
All vector spaces we are going to consider will have as base field Keither Ror C.
Distance functions d on vector spaces E should additionally be translation invariant, i.e. dpx, yq “ dpa `x, a `yq is fulfilled for all x, y, a P E. Then dpx, yq “ dp0, y´xq “: ppy´xq (if we choose a :“ ´x), so d : E ˆE Ñ R is already determined by the mappingp:EÑR.
The triangle inequalitydpx, zq ďdpx, yq `dpy, zqfordtranslates into the subadditivity: ppx`yq ďppxq `ppyq.
Regarding the scalar multiplication we should probably requiredpλx, λyq “λdpx, yq forλą0, i.e.
R`-homogeneity: ppλxq “λ ppxqfor allλPR`:“ ttPR:tą0uandxPE.
Note that this has pp0q “pp2¨0q “2pp0qand hence pp0q “0 as consequence, so also the homogeneitypp0xq “pp0q “0“0ppxqforλ:“0 holds. However, we can not expect the homogeneity for all λPK, because then pwould be linear: In fact,
ppxq `ppyq ěppx`yq “pp´pp´xq ` p´yqqq“ ´ppp´xq ` p´yqq? ě ´ppp´xq `pp´yqq “ppxq `ppyq.
A functionp:EÑRis calledsublinearif it is subadditive andR`-homogeneous.
Note that this is the case if and only if
pp0q “0 andppx`λ¨yq ďppxq `λ ppyq @x, yPE@λą0.
Related to subadditivity is convexity: A functionp:EÑRis calledconvex(see [20,4.1.16]) if
p`
λ x` p1´λqy˘
ďλ ppxq ` p1´λqppyqfor all 0ďλď1 and allx, yPE, so the function lies below each of its chords. By induction this is equivalent to
p`
n
ÿ
i“1
λixi
˘ď
n
ÿ
i“1
λippxiqfor allnPN, xiPE andλią0 with
n
ÿ
i“1
λi“1.
For twice-differentiable functionsf :RÑRone shows in analysis (see [20,4.1.17]) that these are convex if and only iff2ě0 holds:
1.1 Basics 1.2.1 (ð) From f2 ě 0 follows the Mean Value Theorem that f1 is monotonously in- creasing, because f1pxx1q´f1px0q
1´x0 “f2pξq ě0 for some ξ between x0 and x1. So let x0 ă x1, 0 ă λ ă 1 and x “ x0`λpx1´x0q. Again by the Mean Value The- orem, ξ0 P rx0, xs and ξ1 P rx, x1s exist with fpxq ´fpx0q “ f1pξ0q px´x0q and fpx1q ´fpxq “f1pξ1q px1´xq, so
λ fpx1q ` p1´λqfpx0q ´fpxq “
“ p1´λq`
fpx0q ´fpxq˘
`λ`
fpx1q ´fpxq˘
“ p1´λqf1pξ0q px0´xq `λ f1pξ1qpx1´xq
“ p1´λqf1pξ0q`
´λpx1´x0q˘
`λ f1pξ1q`
p1´λq px1´x0q˘
“λp1´λq
´
f1pξ1q ´f1pξ0q
¯
px1´x0q ě0, i.e. f is convex.
(ñ) Letf be convex. Then forx0ăxăx1 withλ:“ xx´x0
1´x0 resp.λ:“xx1´x
1´x0: fpxq ´fpx0q
x´x0 ďfpx1q ´fpx0q
x1´x0 ď fpx1q ´fpxq x1´x . Thus f1px0q ď fpxx1q´fpx0q
1´x0 ď f1px1q, i.e. f1 is increasing monotonously. Thus, we have f2px0q “limx1Œx0 f1pxx1q´f1px0q
1´x0 ě0.
In the definition of “sublinearly” we may replace “subadditive” equivalently by
“convex”:
pðqWe putλ:“ 12 and get ppx`yq “2p
ˆx`y 2
˙ ď2
ˆ1
2ppxq `1 2ppyq
˙
“ppxq `ppyq.
pñqThen p´
λ x` p1´λqy¯
ďppλ xq `ppp1´λqyq “λ ppxq ` p1´λqppyq.
The symmetry dpx, yq “dpy, xqofdtranslates into thesymmetry: ppxq “pp´xq for allxPE. Together with theR`-homogeneity, this is therefore equivalent to the following homogeneity: ppλ xq “ |λ|ppxqforxPE andλPR.
A function p:E ÑR is called seminorm(for short SN) if it is subadditive and positively homogeneous, i.e.ppλ xq “ |λ|ppxqholds forxPE andλPK. A seminorm is therefore a sublinear mapping which fullfills additionally ppλxq “ ppxqfor allxPE and|λ| “1. Note that multiplication with a complex number of absolute value 1 is usually interpreted as a rotation.
Every seminorm pfulfillspě0, because 0“pp0q ďppxq `pp´xq “2ppxq.
A seminorm pis callednormif additionallyppxq “0ñx“0 holds. Anormed space is a vector space together with a norm, cf. [22,5.4.2].
1.2 Important norms
1.2.1 Definition. 8-norm.
Thesupremumor 8-norm is defined by
}f}8:“supt|fpxq|:xPXu,
where f :X ÑKis a bounded function on a setX, cf. [20, 2.2.5].
1.2 Important norms 1.2.4 The distance d, which we looked at in application [18, 1.3] on the vector space CpI,Rq, was just given bydpu1, u2q:“ }u1´u2}8, see also [20,4.2.8]
1.2.2 Examples.
The following vector spaces are normed spaces with respect to the8-norm:
1. For each setX the spaceBpXqof all bounded functionsXÑK;
2. For each compact spaceX the spaceCpXqof all continuous functionsX Ñ K;
3. For each topological space X the space CbpXq of all bounded continuous functions XÑK;
4. For each locally compact spaceXthe spaceC0pXqof all continuous functions X ÑKvanishing at8, i.e. those functions f :X ÑKfor which there is a compact setKĎX for eachεą0, s.t.|fpxq| ăεfor allxRK;
5. If you use (roughly speeking) the maximum of the 8-norms of the deriva- tives, then for each compact manifoldMalso the spaceCnpMqof then-times continuously differentiable functionsM ÑKbecomes a normed space;
On the other hand, we can not use reasonable norms on any of the following spaces:
6. CpXqfor general non-(pseudo-)compactX,
7. The spaceC8pMqof the smooth functions for manifoldsM, 8. CnpMqfor non compact manifoldsM,
9. The space HpGqof holomorphic (i.e., complex differentiable) functions for domainsGĎC.
1.2.3 The variation norm.
Let f : I Ñ K be a function and Z “ t0 “ x0 ă ¨ ¨ ¨ ă xn “ 1u a partition of I“ r0,1s. Then one denotes the variation off onZ by
Vpf,Zq:“
n
ÿ
i“1
|fpxiq ´fpxi´1q|, cf. [22,6.5.11]. The (total) variationof a function is
Vpfq:“sup
Z
Vpf,Zq.
With BVpIqwe denote the space of all functions with bounded variation, i.e.
those functions f for which Vpfq ă 8 holds. It is easy to verify that BVpIqis a vector space, andV is a seminorm onBVpIqwhich vanishes exactly on the constant functions.
1.2.4 Definition. pnorm.
For 1ďpă 8, thep-normis defined by }f}p:“
ˆż
X
|fpxq|pdx
˙1p ,
where |f|p : X Ñ K is an integrable function. For p “ 2 this is a continuous analogue of the Euclidean norm
}x}2:“
g f f e
n
ÿ
i“1
|xi|2
forxPRn orxPCn (here the absolute value in|xi|2is necessary).
1.2 Important norms 1.2.7 The formulaxf|gy:“ş
Xfpxqgpxqdxgeneralizes the inner productx.|.yonKn. Clearly}f g}1ď }f}8¨ }g}1 holds. In order to use the inner product for measureing angles, the inequality of Cauchy-Schwarz }f g}1ď }f}2¨ }g}2 is necessary, see [18, 6.2.1]. A common generalization is the
1.2.5 H¨older inequality.
|xf|gy| ď }f g}1ď }f}p¨ }g}q for 1 p`1
q “1 with1ďp, qď 8 See [23,5.36].
resp.
ż
|f g| ď ˆż
|f|p
˙p1ˆż
|g|q
˙1q
Proof.Let first}f}p“1“ }g}q. Then|fpxqgpxq| ď |fpxq|p p`|gpxq|
q
q , because log is concave (i.e.´log is convex, because log2pxq “ ´x12 ă0) and thus logpa1{p¨b1{qq “
1
ploga`1qlogbďlogp1pa`1qbqfora:“ |fpxq|pandb:“ |gpxq|q, i.e.a1p¨b1q ď 1pa`1qb.
By integration we get }f g}1“
ż
|f g| ď }f}pp p `}g}qq
q “ 1 p`1
q “1.
Let α:“ }f}p and β :“ }g}q be arbitrary (unequal to 0). Then we can apply the first part onf0:“α1f andg0:“ 1βgand get
1
α β}f g}1“ }f0g0}1ď1ñ }f g}1ď }f}p¨ }g}q. The remaining inequality|xf|gy| “ |ş
fg| ď¯ ş
|f| |¯g| “ }f g}1 is obvious.
1.2.6 Minkowski inequality.
}f`g}pď }f}p` }g}p, i.e.} }p is a seminorm See [20,2.2.4], [21,2.72], [23,5.37].
Proof.With 1p `1q “1 we have }f`g}pp“
ż
|f `g|pď ż
|f| |f`g|p´1` ż
|g| |f `g|p´1 ď }f}p¨ }pf`gqp´1}q` }g}p¨ }pf`gqp´1}q
looooooomooooooon
pş
|f`g|pp´1qqq1{q
(H¨older Inequality)
“ p}f}p` }g}pq ¨ }f`g}p{qp sinceq“ p p´1 ñ }f`g}p“ }f`g}pp1´
1 qq
p ď }f}p` }g}p. 1.2.7 Examples.
1. The space CpIqof all continuous functions is a normed space with respect to thep-norm.
2. On the spaceRpIqof all Riemann-integrable functions, however, thep-norm is not a norm but only a seminorm, since a functionf which vanishes except at most finitely many points, nevertheless fulfills}f}p“0.
1.2 Important norms 1.3.3 3. Also`pis a normed space, where`pdenotes the space of sequencesnÞÑxn P
K, which arep-summable, i.e. for whichř8
n“1|xn|p ă 8holds. This space can be identified (via fptq :“ xn for n ď t ă n`1) with left-continuous staircase functions f :tt:tě0u ÑKhaving jumps in at most points inN.
1.3 Elementary properties of seminorms
1.3.1 Lemma. Reverse triangle inequality.
Each seminorm p:EÑRfulfills the reverse triangle inequality:
|ppx1q ´ppx2q| ďppx1´x2q.
Proof.The following applies:
ppx1q ďppx1´x2q `ppx2q ñppx1q ´ppx2q ďppx1´x2q
andpp´xq “ppxq ñppx2q ´ppx1q ďppx2´x1q “ppx1´x2q ñ |ppx1q ´ppx2q| ďppx1´x2q
We now want to give a more geometric description of seminormsp. The idea is to examine the level surfacesp´1pcq.
1.3.2 Definition. Balls.
Letp:EÑRbe a mapping andcPR. Then we put
păc:“ tx:ppxq ăcu and pďc:“ tx:ppxq ďcu,
and call this (ifpis sublinear) theopen and theclosedp-ball around0with radius c. .
1.3.3 Lemma. Balls of sublinear mappings.
For each sublinear mapping 0 ďp : E Ñ R and c ą 0, pďc and păc are convex absorbing subsets ofE. We havepďc“c¨pď1 as well aspăc“c¨pă1, and further ppxq “c¨inftλą0 :xPλ¨pďcu.
So we may recover the mappingpfrom the unit ballpď1. A set A Ď E is called convex (see [22, 5.5.17]), if řn
i“1λixi P A follows from λi ě0 withřn
i“1λi “1 andxiPA. It suffices to asssume this for n“2, because fornă2 it is obvious and fromn“2 it follows for allną2 by induction:
n`1
ÿ
i“1
λixi“λn`1xn`1` p1´λn`1q
n
ÿ
i“1
λi
1´λn`1xi. A setAis called absorbentif@xPEDλą0 :xPλ¨A.
Proof.Forcą0 we have:
pďc “ tx:ppxq ďcu “!
x:p´x c
¯
“ 1
cppxq ď1)
“ tc y:ppyq ď1u “c¨ ty:ppyq ď1u “c¨pď1
and analogously forpăc.
The convexity of pďc “p´1tλ : λ ď cu and păc “ p´1tλ : λ ă cu immediately follows from the easy-to-see property that inverse images of intervals, being un- bounded from below, under convex functions are convex.
1.3 Elementary properties of seminorms 1.3.6 To see that pďc “c¨pď1 is absorbent for cą0, it is sufficient to put c“1: Let x P E be arbitrary. If ppxq “ 0, then x P pď1. Otherwise, x P ppxq ¨pď1 holds because x“ppxq ¨y, wherey:“ ppxq1 xandppyq “ppppxq1 xq “ppxq1 ppxq “1.
Hence also the supersetpăcĚpďc{2is absorbent.
Because of following equivalences forλą0 we haveppxq “inftλą0 :xPλ¨pď1u:
xPλ¨pď1“pďλôppxq ďλ, hence
inftλą0 :xPλ pď1u “inftλą0 :λěppxqu “ppxq.
1.3.4 Lemma. Balls of seminorms.
For each seminormp:EÑRandcą0,păc andpďc are absorbent and absolutely convex and
ppxq “inf
!
λą0 :xPλ¨pď1“pďλ
) .
A subset A ĎE is called balanced, if for all xP A and |λ| “ 1 also λ¨xP A holds.
More generally, a subset A ĎE is called absolutely convexif it follows from xiPAandλiPKwithřn
i“1|λi| “1 thatřn
i“1λixiPA holds.
Sublemma.
A set A is absolutely convex if and only if it is convex and balanced.
Proof.pñqis clear, because every convex combination is also an absolutely convex combination and for |λ| “ 1 also λ x is an absolutely convex combination. Note that for this it is sufficient to have absolutely convexity forn“2, because that for n“1 it follows fromλ1x1“λ1x1`0x1.
pðqLetřn
i“1|λi| “1, then
n
ÿ
i“1
λixi “ ÿ
λi‰0
λixi“ ÿ
λi‰0
|λi| λi
|λi|xi PA, holds because of
ˇ ˇ ˇ
λi
|λi|
ˇ ˇ
ˇ“1 and therefore, because of the balancedness |λλi
i|xi PA, and therefore, because of the convexity, also ř
λi‰0|λi||λλi
i|xi PAholds.
This proof shows that even for “absolutely convex” it is enough to ask this for the casen“2.
Proof of the lemma 1.3.4 .Because of the previous lemma and the sublemma, only balancing is to be shown, and this is obvious because of the positive homo- geneity ofp.
1.3.5 Definition. Minkowski functional.
We now want to construct from setsArelated seminormsp. For this we define the Minkowski functionalpA:
xÞÑpApxq:“inftλą0 :xPλ¨Au PRY t`8ufor eachxPE.
ThenpApxq ă 8holds if and only ifxlies in the conetλPR:λą0u ¨Agenerated byA.
1.3.6 Lemma. From balls to seminorms.
1.3 Elementary properties of seminorms 1.3.7 LetAbe convex and absorbent. Then the Minkowski functional ofAis a well-defined sublinear mapping p:“pAě0 onE, and for λą0 we have:
păλĎλ¨AĎpďλ. If A is also absolutely convex, thenpis a seminorm.
So we can recover the setA almost from the function p.
Proof.SinceAis absorbent, the cone istλ:λą0u ¨A“E. Sopis finite onE.
Furthermore, 0PAholds, becauseDλą0 : 0Pλ A and thus 0“ 0λ PAholds.
The function pisR`-homogeneous, because forλą0 we have:
ppλxq “inftµą0 :λ xPµ Au
“inf
!
µą0 :xP µ λA
)
“inftλ νą0 :xPν Au “λinftν ą0 :xPνAu
“λ ppxq.
ppăλĎλ¨AqLetppxq “inftµą0 :xPµ Au ăλ. Then there is a 0ăµďλwith x Pµ A“λµλA Ďλ A, because 0P A and thus µλa “ p1´µλq0`µλa PA for all aPA.
pλ¨AĎpďλqIfxPλ A, then by definition ofpit is clear thatppxq ďλ, i.e.xPpďλ. The function pis subadditive because
ppxq ăλ, ppyq ăµñxPλ A, yPµ A
ñx`yPλ A`µ A“ pλ! `µqAñppx`yq ďλ`µ ñppx`yq ďinftλ`µ:ppxq ăλ, ppyq ăµu “ppxq `ppyq, holds, since for convex sets A and λi ą 0 we have řn
i“1λiA “ přn
i“1λiqA: In fact, xi P A implies ř
iλixi “ ř
iλ¨ λλixi “ λ¨ř
i λi
λ xi P př
iλiq ¨A, where λ:“řn
i“1λi, and thusř
i λi
λ xi is a convex combination. Conversely,xPAimplies přn
i“1λiqx“ř
iλixPř
iλiA.
IfA is additionally absolutely convex thenpis a seminorm, because ppλ xq “ppxq holds for all|λ| “1 sinceAis balanced, soλ A“Ais fullfilled.
1.3.7 Lemma. Comparison of seminorms.
For each two sublinear mappings p, qě0:
pďqôpď1Ěqď1ôpă1Ěqă1. Proof.p1ñ3qThe following holds:
xPqă1ñppxq ďqpxq ă1ñxPpă1. p3ñ2qThe following holds:
xPqď1ñqpxq ď1 ñ @λą1 :q
´x λ
¯
“ 1
λqpxq ď 1 λ1ă1 ñ x
λ Pqă1Ďpă1ñ 1
λppxq “p
´x λ
¯
ă1ñppxq ăλ ñppxq ďinftλ:λą1u “1
ñxPpď1
1.3 Elementary properties of seminorms 1.4.1 p2ñ1qThe following holds:
Letλą0 be s.t. 0ďqpxq ăλñq
´x λ
¯
“ 1
λqpxq ďλ λ “1 ñ x
λPqď1Ďpď1 ñp
´x λ
¯
ď1, i.e.ppxq ďλ ñppxq ďinftλ:λąqpxqu “qpxq
1.4 Seminorms versus topology
1.4.1 Topologies generated by seminorms.
Motivation: The seminorms provide us, as in Analysis, with balls, which we want to use for questions of convergence and continuity. For this the notion of a topology has been developed:
In Analysis, we callOĎRopen if there is anδ-neighborhoodU ĎOfor eachaPO (i.e. a setU :“ tx:|x´a| ăδuwithδą0).
This definition can be transfered almost literally to normed spaces pE, pq:
OĎE is calledopen :ô @aPO Dδą0:tx:ppx´aq ăδu ĎO. Note that tx:ppx´aq ăδu “a`păδ “a`δ¨pă1,
because ppx´aq ăδôx“a`y withy:“x´aPpăδ.
But important function spaces do not have a reasonable norm. For example, we can no longer consider the supremum norm on CpR,Rq. But for each compact interval KĎRwe may consider the supremumpK onK, i.e.pKpfq:“supt|fpxq|:xPKu.
We call OĎE open with respect to a given family P0 of seminorms on a vector spaceE, if
@aPODnPNDp1, . . . , pn PP0, Dεą0 :tx:pipx´aq ăεfori“1, . . . , nu ĎO.
The family O:“ tO:OĎE ist openudefines then a topology onE, the so-called topology generated by P0 (unions of the so defined open sets are obviously open again and the same applies for intersections of finitely many open sets, because the union of finitely many sets, each consists of finite many seminorms, is finite and the minimimum of the finitely manyεą0 is positive). Generally, atopology(see [26,1.1.1])Oon a setX is a setOof subsets ofX, which fullfills the following two conditions:
1. IfF ĎO, then the unionŤ F“Ť
OPFO belongs toO;
2. IfF ĎO is finite, the intersectionŞ F“Ş
OPFO is also inO.
Note thatŤ
H “ H andŞ
H:“X. The subsetsO ofX, which belong toO, are also called open setsof the topology in the general case. A topological space is a set together with a topology.
The above construction is a general principle. One calls a subsetO0ĎOsubbasis of a topology O, if @aPO PO DF ĎO0, finite: aPŞ
F ĎO, cf. [26,1.1.6].
In order to construct a topology O it is sufficient to specify a set O0 of subsets of X, and then to designateO as the set of all O ĎX for which there is a finite subsetF ĂO0 withxPŞ
F ĎO for each of the pointsxPO. One says, that the topology Ois generated by the sub-basisO0.
1.4 Seminorms versus topology 1.4.2 The topology generated by P0 is just the topology generated by sub-basis O0 :“
ta`păε:aPE, pPP0, εą0u.
(Ď) The topology generated byP0 is obviously coarser or equal to that generated by the sub-basisO0, because all we have to do is to set allai“aand εi “ε.
(Ě) In fact, letO ĎE be open in the latter topology, i.e.@aPO DF ĎO0,finite:
aPŞ
F ĎO. SoDa1, . . . , an PE,p1, . . . , pnPP0 andε1, . . . , εną0 with aP txPE:pipx´aiq ăεi fori“1, . . . , nu ĎO.
If we put now ε:“mintεi´pipa´aiq:i“1, . . . , nu, i.e.
aP txPE:pipx´aq ăεfori“1, . . . , nu
Ď txPE:pipx´aiq ďpipx´aq `pipa´aiq ăεi fori“1, . . . , nu ĎO.
By a neighborhood U of a point ain a topological space X, one understands a subsetU ĎX for which an open setOPO exists withaPOĎU.
Aneighborhood(sub)basisU of a pointain a topological spaceX is a setU of neighborhoodsU ofasuch that for each neighborhoodO, a set (finitely many sets) UiPU exists (exist), so thatŞ
iUiĎO, cf. [26,1.1.7].
As in Analysis, a mapping f : X ÑY between topological spaces is called con- tinuous at aPX, if the inverse image of each neighborhood (in a neighborhood basis) offpaqthere is a neighborhood ofa, cf. [26,1.2.4]. It is called continuous, if it is continuous in each pointaPX, that is the case if and only if the inverse image of each open set is open. It is easy to see that it is sufficient to check this condition for the elements of a sub-basis.
Each seminorm p P P0 is continuous for the topology generated by P0, because if a P E and ε ą 0, then ppa`păεq Ď tt : |t´ppaq| ă εu, since x P păε ñ
|ppa`xq ´ppaq| ďppxq ăε. But also the addition`:EˆEÑE is continuous, because pa1`păεq ` pa2`păεq Ď pa1`a2q `pă2ε. In particular, the translations xÞÑa`xare homeomorphisms.
The scalar multiplication ¨ : KˆE Ñ E is continuous. For λ P K and a P E:
tµPK:|µ´λ| ăδ1u ¨ tx:ppx´aq ăδ2u Ď tz:ppz´λ¨aq ăεuifδ1ă 2ppaqε and δ2ă ε2p|λ| `2ppaqε q´1, since
ppµ¨x´λ¨aq “pppµ´λq ¨x`λ¨ px´aqq ď |µ´λ| ¨ppxq ` |λ| ¨ppx´aq ďδ1¨ pppaq `ppx´aqq ` |λ| ¨δ2
ďδ1¨ pppaq `δ2q ` |λ| ¨δ2“δ1¨ppaq `δ2¨ pδ1` |λ|q ď ε
2 `ε 2
ˆ
|λ| ` ε 2ppaq
˙´1
¨ ˆ
|λ| ` ε 2ppaq
˙
“ε.
In particular, the homotheticsxÞÑλ¨xare homeomorphisms forλ‰0.
So the topology generated byP0 turnsE into atopological vector space, i.e.
a vector space together with a topology with respect to which the addition and the scalar multiplication are continuous. Moreover, E is even a locally convex vector space, i.e. there exists a 0-neighborhood basis consisting of (absolutely) convex sets (namely,Şn
i“1ppiqăε), or a sub-basis consisting of (absolutely) convex sets (namely,păε).
1.4.2 Lemma. Continuity of seminorms.
1. A seminormp:EÑRon a topological vector spaceE is continuous if and only ifpă1 (or, equivalently,pď1) is a 0-neighborhood.
1.4 Seminorms versus topology 1.4.4 2. A seminormp:EÑRis continuous in the topology generated byP0 if and
only ifDp1, . . . , pnPP0, λą0:pďλ¨maxtp1, . . . , pnu.
Proof.
1 (ñ) Sincepis continuous, 0Pp´1tt:tă1u “pă1is open.
(ð)
aPa`ε¨pă1“ tx:ppx´aq ăεu Ďp´1tt:|t´ppaq| ăεu.
2 (ñ) If p is continuous, then pă1 is a 0-neighborhood, so p1, . . . , pn P P0 and εą0 exist with
pă1Ě
n
č
i“1
ppiqăε“
n
č
i“1
εppiqă1“ε
n
č
i“1
ppiqă1“εpmaxtp1, . . . , pnuqă1
“ pmaxtp1, . . . , pnuqăε“qă1,
where q:“ 1ε¨maxtp1, . . . , pnu. Thuspďq:“1ε¨maxtp1, . . . , pnuholds by 1.3.7. (ð) With pi alsoq :“λ¨maxtp1, . . . , pnu is continuous, and thuspă1Ěqă1 is a 0-neighborhood, i.e.pcontinuous by 1 .
1.4.3 Summary.
Let P0 be a family of seminorms on a vector space E. Then the balls a`păε :“
txPE :ppx´aq ăεuwith pPP0,εą0 andaPE form a sub-basis of a locally convex topology. This so-called topology generated by P0 is the coarsest topology (i.e. with the fewest open sets) on E, for which all seminormspPP0as well as all translations xÞÑ a`x with a PE are continuous. With respect to this topology, a seminorm p on E is continuous if and only if there are finite many seminorms piPP0 and oneKą0, s.t.
pďKmaxtp1, . . . , pnu.
1.4.4 Definition. Seminormed space.
By aseminormed spacewe therefore understand a vector spaceE together with a set P of seminorms, which are just the continuous seminorms of the topology generated by it, that is, withp1, p2PP also every seminormpďp1`p2 is inP. A setP0ĎPis calledsub-basis of the seminormed spacepE,Pq, if it generates the same topology asP, that is for any seminormpinP finite manyp1, . . . , pnPP0 exist as well as aλą0 withpďλ¨maxtp1, . . . , pnu.
For any family P0 of seminorms onE, we get a uniquely determined seminormed space, which has P0 as sub-basis of its seminorms, by using the familyP of, with respect to the topology generated byP0, continuous seminorms:
P :“
!
pis a seminorm onE:Dλą0Dp1, . . . , pnPP0 withpďλ¨maxtp1, . . . , pnu )
. By theseminorms of the so obtained seminormed space we understand all seminorms belonging to the generating family P0. We would actually have to say
“seminorms of the given sub-basis of the seminormed space”, but that’s too long for us.
By a countably seminormed space we mean a seminormed space which has a countable sub-basisP0 of seminorms. We may then assume thatP0“ tpn :nPNu and the sequence ppnqn is monotone increasing and will eventually dominate any continuous seminorm p, that is there is an n P N with p ď pn. To achieve this, replace the pn withn¨maxtp1, . . . , pnu.
1.4 Seminorms versus topology 1.4.9 1.4.5 Definition. Convex hull.
The convex hull xAykv of a subset A Ď E is the smallest convex subset of E which includes A.
1.4.6 Lemma. Convex hull.
Let AĎE. Then the convex hull ofA exists and is given by xAykv “č
tK:AĎKĎE, K is convexu
“
!ÿn
i“1
λiai:nPN, aiPA, λiě0,
n
ÿ
i“1
λi“1 )
.
Proof.The setA:“ tK:AĎKĎE, K ist convexuis not empty, becauseEPA.
Consequently there existsŞ
Aand obviously is itself convex and thus the minimal element inA, i.e.xAykv“Ş
A.
For the second description of the convex hull note that the setA0:“ třn
i“1λiai: n PN, ai PA, λi ě0,řn
i“1λi “1uobviously includes A. It is convex, because let xj PA0, i.e. xj “řnj
i“1λi,jai,j fornj PN, ai,j P A, λi,j ě0 with řnj
i“1λi,j “1.
Then forµj ě0 withřm
j“1µj “1 we have:
m
ÿ
j“1
µjxj“
m
ÿ
j“1
µj nj
ÿ
i“1
λi,jai,j “ ÿ
i,j iďnj
µjλi,jai,j
with ÿ
iďnj
µjλi,j“
m
ÿ
j“1
µj
nj
ÿ
i“1
λi,j“
m
ÿ
j“1
µj1“1.
SinceA0is clearly contained in every setKPA,xAykv“A0 holds.
1.4.7 Definition. Absolutely-convex hull.
Theabsolutely convex hullxAyakvof a subsetAĎEis the smallest absolutely convex subset ofE that containsA, thus is the intersection of all these sets.
1.4.8 Lemma. Absolutely-convex hull.
Let AĎE. Then the absolutely convex hull is given by xAyakv“ xtλ:|λ| “1u ¨Aykv, so it is the convex hull of the balanced hull tλ:|λ| “1u ¨A.
Proof. It is only to be shown that the convex hull of a balanced set A is itself balanced. So let|µ| “1 andřn
i“1λiaiP xAykv, then µ¨
n
ÿ
i“1
λiai“
n
ÿ
i“1
λiµ aiP xAykv, sinceµ¨aiPA.
1.4.9 Lemma.
Each locally convex vector space E has a 0-neighborhood base of absolutely convex sets.
Proof. Let U be a convex 0-neighborhood. This is open without restriction of generality, because its interior is also convex(!). Since the scalar multiplicationtλP K:|λ| “1u ˆEÑEis continuous and 0¨λ“0 holds, there exists a neighborhood VλĎKofλfor each|λ| “1 and a convex 0-neighborhoodUλĎEwithVλ¨UλĎU.
1.4 Seminorms versus topology 1.5.2 Since tλ P K: |λ| “ 1u is compact, finitely many exist λ1, . . . , λn with tλ P K:
|λ| “1u ĎŤn
i“1Vλi. LetU0:“Şn
i“1Uλi. ThenU0 is a convex 0-neighborhood and U0ĎU1 :“ tλPK:|λ| “1u ¨U0 ĎU. The convex hull of the balanced set U1 is thus an absolutely convex 0-neighborhood in U by 1.4.8.
1.4.10 Remarks.
The topology of each locally convex vector space is generated by the set P of all continuous seminorms:
(Ě) If O is open in the topology generated by P, then for every a P O finitely many p1, . . . , pn PP andεą0 exist withŞn
i“1
`a`ε¨ ppiqă1
˘“ x:pipx´aq ă ε@i“1, . . . , n(
ĎO, soOis also in the original topology open since theppiqă1are 0-neighborhoods.
(Ď) Conversely, let the latter be fulfilled, i.e. by 1.4.9 there exists an absolutely convex 0-neighborhood U with U Ď O ´a for each a P O. Then p :“ pU is a continuous seminorm, because pď1 Ě U is also a 0-neighborhood. Consequently, a`pă1 Ďa`U Ď O holds, so O is also open in the topology generated by the continuous seminorms.
Since we only have to use the Minkowski functionals of a 0-neighborhood basis in this argument, the following holds:
The topology of each locally convex vector space is already generated by the Minkowski functionals of a 0-neighborhood basis consisting of absolutely convex sets.
1.4.11 Corollary. Special 0-neighborhood basis.
Each locally convex vector space E has a 0-neighborhood basis consisting of closed absolutely convex sets.
Proof.This is obvious becauseppUqď1{2ĎU is closed.
1.4.12 Summary.
Let E be a locally convex vector space andU a 0-neighborhood sub-basis consisting of absolutely convex sets. Then the family tpU : U P Uu is a sub-basis of that seminormed space, whose seminorms are exactly those being continuous with respect to the given topology, these are exactly those seminorms q for which qď1 is a 0- neighborhood.
So we have a bijection between seminormed spaces and locally convex vector spaces, and can work with topology or with seminorms on a fixed vector space as needed.
1.5 Convergence and continuity
1.5.1 Definition. Convergent sequence.
A sequence pxiqi convergestowardsain a topological spaceX if and only if for each neighborhoodU (of a sub-basis) ofaan indexiU exists, such thatxiPU for alliěiU, cf. [26,1.1.11].
1.5.2 Lemma. Convergent sequences.
A sequence pxiqconverges in the underlying topology of a locally convex space with sub-basis P0 towards aif and only if ppxi´aq Ñ0 for allpPP0.
1.5 Convergence and continuity 1.5.6 Proof. pñq Since for a P E the translation y ÞÑ y´a is continuous, xi´a Ñ a´a“0, and thus alsoppxi´aq Ñpp0q “0 for each continuous seminormp.
pðqLetU be a neighborhood ofa. Then there are finitely many seminormspjPP0
and aεą0 witha`Şn
j“1ppjqăεĎU. Sincepjpxi´aq Ñ0, for eachjthere exists anij withpjpxi´aq ăεforiěij. LetI be greater than all the finitely manyij. ThenxiPa`Şn
j“1ppjqăεforiěI and thus also inU, i.e.xiÑa.
1.5.3 Lemma. Sequentially continuous mapping.
A mapping f :EÑX of a countably seminormed spaceE into a topological space X is continuous if and only if it is sequentially continuous, i.e. for each convergent sequence xiÑaalso the image sequence fpxiq Ñfpaqconverges.
See [20,3.1.3].
Proof. pñq is clear, because of the above description 1.5.2 of the convergent sequences.
pðqindirectly: Suppose f´1pUqis not a neighborhood of afor a neighborhood U offpaq. Lettpn:nPNube a countable sub-basis of the seminorms ofE. Then for each n there is anxn P E with pkpxn´aq ă n1 for all k ďn and fpxnq R U. So pkpxn´aq Ñ0 for nÑ 8, and thus also xn Ñ aaccording to the above lemma 1.5.2. But sincefpxnq RU, this is a contradiction to the sequential continuity of f.
1.5.4 Definition. Net.
Since the above lemma does not hold for non-countably seminormed spaces , we extend the notion of a sequence to:
A net (generalized sequence or Moore-Smith sequence, see [26,3.4.1]) is a mapping x: I ÑX, where I is a directedindex set, i.e. a set together with a relation ă, which is transitive and has for any two elementsi1 andi2 in I also a iP I with i1 ăi and i2 ăi, see also [26, 3.4.1]. Exactly, as for sequences, one defines the convergence of nets and shows thus also the first of the two lemmas from above. Regarding the second lemma we have
1.5.5 Lemma. Continuity via nets.
A mappingf :EÑX from a locally convex space to a topological space is contin- uous if and only if for each convergent netxiÑathe image netfpxiq Ñfpaq. See [26,3.4.3].
Proof. pñq is obvious, because if U is a fpaq-neighborhood and xi Ñ a, then Di0@iěi0:xiPf´1pUq, i.e.fpxiq PU, that isfpxiq Ñfpaq.
pðqLetU be a neighborhood basis ofa. Then we use as index setI:“ tpU, uq:U P U, uPUuwith the orderpU, uqăpU1, u1q ôU ĚU1 and as net on it the mapping x : pU, uq ÞÑ u. Then, clearly, the net x converges to a, so by assumption also f ˝xtowardsfpaq, i.e. for eachfpaq-neighborhoodV exists an indexpU0, u0q, s.t.
fpuq PV for allU ĎU0anduPU. SofpU0q ĎV, that meansf is continuous.
1.5.6 Definition. Separatedness.
A locally convex space is called separated (or also Hausdorff, see [26,3.4.4]), if the limits of convergent sequences (or nets) are unique, this is the case if and only ifppxq “0 for allpPP0 impliesx“0:
1.5 Convergence and continuity 1.6.3 pðqLetxi be a net converging tox1 andx2. Thenxi´x1 converges towards 0 and also towardsx2´x1. Because of the continuity ofp,ppxi´x1qconverges topp0q “0 and also toppx2´x1q. Because of the uniqueness of the limits inK,ppx2´x1q “0 holds for allp, and thus, by assumption,x2´x1“0.
pñqLetppxq “0 for allp. Then the constant sequence (net) with valuexconverges to both 0 and x, hence, by assumption,x“0.
We are going to use the abbreviation lcsfor separated locally convex spaces.
1.6 Normable spaces
1.6.1 Definition. Normable spaces and bounded sets.
One calls a separated lcs, which has a sub-basis consisting of a single (semi-)norm, normable.
A set B ĎE is calledbounded if and only if ppBq is bounded for all pPP0, cf.
[20,2.2.9]. That’s exactly the case when it gets absorbed by all 0-neighborhoods, i.e. @0-neighborhoodU DKą0 :BĎK¨U:
pðqLetpbe a continuous seminorm, thenpď1is a 0-neighborhood, so by assump- tion there is an K ą 0 with B Ď K ¨pď1 “ pďK, i.e. p is bounded on B by K.
pñqLet U be a 0-neighborhood. Then there are finitely many seminormspi PP0
and anεą0 withŞn
i“1ppiqďεĎU. For eachpithere is aKią0 with|pipBq| ďKi, so BĎŞn
i“1ppiqďKiĎŞn
i“1ppiqďK ε“K¨U, whereK:“1ε¨maxtK1, . . . , Knu.
1.6.2 Theorem of Kolmogoroff.
A separated lcs is normable if and only if it has a bounded zero-neighborhood.
Proof. pñq Let p be a norm generating the structure. Then U :“ pď1 is a 0- neighborhood. For any continuous seminormqthere exists anKą0 withqďK¨p, and thusq is bounded onU byK. SoU is bounded.
pðqLetU be a bounded zero neighborhood. Then there is a continuous seminorm with pď1ĎU. Now letq be any seminorm. SinceU is bounded, there is a K ą0 with |qpUq| ď K. So pď1 Ď U Ď qďK “ pK1qqď1 and therefore p ě K1q, that is qďK¨p. Thus,tpuis a sub-basis of the seminorms ofE andpis even a norm.
1.6.3 Example. The pointwise convergence of continuous functions.
The pointwise convergence on CpI,Rq can not be a normed space.
Proof.A sub-basis of seminorms for pointwise convergence is given byf ÞÑ |fpxq|
for x P I. Suppose there is a bounded zero neighborhood B. Then finitely many points x1, . . . xn PI and a εą0 exist, s.t. B :“ tf : |fpxiq| ăεfori“1, . . . , nu is bounded. Let x0 R tx1, . . . , xnu. Then the seminorm q : f ÞÑ |fpx0q| is not bounded on B, because certainly there exists a (polynomial) f which vanishes on tx1, . . . , xnu, but not onx0, and thusK¨f PB, butqpK¨fq “K¨fpx0q Ñ 8 for KÑ 8.
Analogously one shows that the uniform convergence on compact sets in the space CpR,Rq is not normable but yields a countably seminormed space. And similarly for the uniform convergence in each derivative onC8pI,Rq.
2. Linear mappings and completeness
In this chapter we examine the basic properties of linear mappings as well as the notion of completeness and its relevance for power series. In particular, we apply this to prove the inverse function theorem and the Weierstrass approximation theorem, as well as for solving linear differential equations.
2.1 Continuous and bounded mappings
2.1.1 Lemma. Continuity of linear mappings.
For a linear mapping f :EÑF between lcs’s are equivalent:
1. f is continuous;
ô2. f is continuous at 0;
ô3. For each (continuous) SN q ofF,q˝f is a continuous SN of E.
Proof.p1 ñ 3qqa continuous SN,f continuous linear ñq˝f is a continuous SN.
p3 ñ 2q Let U be a 0-neighborhood of 0 “ fp0q in F, without restriction of generality U “Ş
ity :qipyq ăεufor SN’s q1, . . . , qn ofF. Thenf´1pUq “ Ş
itx: qipfpxqq ăεu “Ş
ipqi˝fqăεis open inE.
p2 ñ 1q We have fpxq “ fpx´aq `fpaq, i.e. f “Tfpaq˝f ˝T´a, where the translations T´a and Tfpaq are continuous and the middle f is continuous at 0, hence also the compositionf is continuous atpT´aq´1p0q “a.
2.1.2 Lemma. Continuity of multi-linear mappings.
An n-linear mappingf :E1ˆ. . .ˆEnÑF between lcs’s is continuous if and only if it is continuous at 0.
Proof.Let firstn“2. ForaiPEiand any neighborhoodfpa1, a2q `W offpa1, a2q with absolutely convexW, 0-neighborhoodsUi exist inEiwithfpU1ˆU2q Ď13W, because of the continuity of f at 0. Now choose a 0 ă ρ ă1 with ρ ai P Ui for i“1,2. Thenfppa1`ρ U1q ˆ pa2`ρ U2qq Ďfpa1, a2q `W, becauseuiPUi is
f`
a1`ρ u1, a2`ρ u2
˘´f` a1, a2
˘“fpa1, ρ u2q looooomooooon
“fpρ a1,u2q
`fpρ u1, a2q looooomooooon
“fpu1,ρ a2q
`fpρ u1, ρ u2q loooooomoooooon
“ρ2fpu1,u2q
Ď 1 3W`1
3W `1
3W ĎW.
Forną2, chooseU1, . . . , Un analogously withp2n´1qfpU1ˆ. . .ˆUnq ĎW. 2.1.3 Definition. Bounded linear mappings.
A linear mapping is calledboundedif the image of each bounded set is bounded.
Warning: In the literature this notation is sometimes also used for the non-equivalent
2.1 Continuous and bounded mappings 2.1.6 property to be bounded on some 0-neighborhood!
Note that bounded subsets of an LCS can not contain any raya`R`¨v forv‰0, since otherwise tÞÑppa`t vqwould be bounded on R`, say byKp ą0, for each seminormpofE, hencet ppvq “pptvq ďppa`tvq `pp´aq ďKp`ppaqfor alltą0 by 1.3.1, henceppvq “0, i.e.v“0.
Consequently, a linear mapping f :EÑF is bounded as mapping from the setE toF (i.e.fpEq ĎF is bounded), only if is the 0-map, becausefpEqwould then be a bounded linear subspace, and thusfpEq “ t0u.
2.1.4 Lemma. Bounded linear mappings.
For linear mappings f :EÑF between lcs’s the following implications hold:
1. f is continuous;
ñ2. f is sequentially continuous;
ñ3. f is bounded.
Proof.(1ñ 2 ) holds even for non-linearf by 1.5.5.
(2 ñ 3) SupposefpBqis not bounded for some bounded setBĎE. Then there is a seminorm q of F and a sequence bn P B, s.t. 0 ă λn :“ qpfpbnqq Ñ 8.
The sequence λ1
nbn then converges to 0 (see the following lemma), so because of the sequential continuity also fpλ1
nbnq “ λ1
nfpbnq and thus also qpλ1
nfpbnqq “
1
λnqpfpbnqq “1, a contradiction.
Now the question arises of the validity of the converse to the implications in 2.1.4. Forp1ð2qwe have already answered this positively in 1.5.3 for countably semi- normed spaces.
Forp2ð3qwe need some relationship between bounded and convergent sequences.
A simple fact is the following.
2.1.5 Lemma. Mackey-convergence.
Let tyn :nPNu ĎE be bounded in an lcs andρn Ñ0 inR. ThenρnynÑ0.
Proof.By applying seminorms this is reduced to the corresponding result forR. Or directly: LetU be an absolutely convex 0-neighborhood. Thentyn:nPNu ĎK¨U for some Ką0 and thus ρnynPU for all|ρn| ď K1, so for almost alln.
In order to be able to deduce at least sequential continuity from boundedness, it would be helpfull if the converse were true, i.e. if we could write any convergent sequencepxnqninEas a product of a bounded sequencepynqninEand a 0-sequence ρn inR. A sequencepxnq, for which this holds, is calledMackey 0-sequenceor Mackey-convergent towards 0, so if D0 ď λn Ñ 8, s.t. tλnxn : n P Nu is bounded.
Each Mackey 0-sequence pxnqn converges to 0 by Lemma 2.1.5 applied toyn :“
λnxn. For normable spaces, the converse implication also holds, because xn Ñ 0 implies 0 ď λn Ñ 8, whereλn :“ }x1
n} for xn ‰ 0 and λn :“n otherwise, and obviously tλnxn : n PNu is bounded in the norm by 1. More generally, this also holds for countably seminormed spaces:
2.1.6 Lemma.
In countably seminormed spaces E, each sequence converging to 0 is even Mackey- convergent to 0.
2.1 Continuous and bounded mappings 2.1.8 Proof.Lettpk:kPNube a monotonously increasing sub-basis ofE andxnÑ0 a 0-sequence. The idea is to define for the countable many zero sequences ppkpxnqqn
forkPNanother zero sequencenÞÑ λ1
n ą0 converging slower towards 0.
0 1
12 13
n1 n2 n3
p1 p2 p3
p1 p2 p3 1Λ
1Λ 1Λ
From pkpxnq Ñ 0 for n Ñ 8 follows the existence of nk P N with pipxnq ď
1
k for all n ě nk and all i ď k. Without loss of generality k ÞÑ nk is strictly monotonously increasing. We define λn :“ k for nk`1 ą n ě nk. Then, n ÞÑ λn
is monotonously increasing, λn Ñ 8, and for n ě nk, pkpλnxnq “ λnpkpxnq “ j pkpxnq ďj pjpxnq ďj1j “1, wherejěkis selected to benj`1ąněnj. 2.1.7 Corollary. Bornologicity of metrizable lcs.
Every countably seminormed space is bornological. Even more holds: Multilinear bounded mappings on countably seminormed spaces are continuous.
Where an lcs is called bornological, if each bounded linear mapping on it is continuous.
In 4.2.5 we will give examples of lcs’s that are not bornological.
Proof. Because of 1.5.3, we only need to show the sequential continuity (at 0) of each boundedm-linear mappingf. Let xn Ñ0. By Lemma 2.1.6 there exists a sequence λn Ñ 8, so thatλnxn is bounded. Then, by assumption fpλnxnq “ λmn fpxnqis also bounded, and thusfpxnqis a (Mackey) 0-sequence by 2.1.5. 2.1.8 Lemma. Continuity in normed spaces.
For linear mappings f :EÑF between normed spaces are equivalent:
1. f is continuous;
ô2. f isLipschitz, i.e.DKą0 :}fpxq ´fpyq} ďK¨ }x´y};
ô3. }f} ă 8.
The operator norm||f||on f is defined as follows (cf. [22,5.4.10]) }f}:“sup }fpxq}:}x} ď1(
“sup }fpxq}:}x} “1(
“sup
!}fpxq}
}x} :x‰0 )
“inf
!
K:}fpxq} ďK}x}for all x )
If f is multi-linear, thenf is continuous if and only if }f}:“sup
"
}fpx1, . . . , xnq}
}x1}. . .}xn} :xi‰0
* ă 8.
2.1 Continuous and bounded mappings 2.2.1
Proof.p1 ô 3 qf is continuous 2.1.7
ô f is bounded on bounded sets (without restriction of generality on tx : }x} ď 1u, since fpBq Ď c¨fptx : }x} ď 1uq for B Ďc¨ tx:}x} ď1u)ôsupt}fpxq}:}x} ď1u “:}f} ă 8.
The following applies:
supt}f x}:}x} “1u ďsupt}f x}:}x} ď1u (because more elements) ďsup
"
}f x}
}x} :x‰0
*
(because}f x} ď }f x}
}x} for||x|| ď1q ďsupt}f x}:}x} “1u (because }f x}
}x} “ }fp 1 }x}xq}q, so equality holds everywhere. Furthermore:
inf
!
K:}f x} ďK¨ }x}for allx )
“inf
"
K: }f x}
}x} ďK for allx‰0
*
“inf
"
K: sup
"
}f x}
}x} :x‰0
* ďK
*
“sup
"
}f x}
}x} :x‰0
* .
The mapping f is Lipschitzô
!}f z}
}z} :z‰0 )
“
!}f x´f y}
}x´y} :x‰y )
is bounded.
The statement for multilinear mappings f is shown analogously.
2.1.9 Corollary. Operator norm.
Let E andF be normed spaces, then the set
LpE, Fq:“ tf :EÑF|f is linear and boundedu
is a normed space with respect to the pointwise vector operations and the operator norm as defined in 2.1.8. Furthermore: }idE} “1 and}f ˝g} ď }f} ¨ }g}.
Proof.The following applies:
@x:}pf`gqx} ď }f x} ` }gx} ď p}f} ` }g}q }x} ñ }f`g} ď }f} ` }g}
@x:}pλ fqx} “ |λ| }f x} ñ }λ f} “ |λ| }f}
@x:}pf˝gqx} ď }f} }g} }x} ñ }f˝g} ď }f} }g}.
Attention}f ˝g} ‰ }f} ¨ }g}, e.g.fpx, yq:“ px,0qandgpx, yq:“ p0, yq.
2.1.10 Definition. Normed algebra.
Anormed algebrais a normed spaceAalong with a bilinear mapping‚:AˆAÑ A, which is associative, has a unit 1 and satisfies}1} “1 as well as}a‚b} ď }a} ¨ }b}.
One of the most important examples isLpE, Eq “:LpEqfor normed spacesE.
2.2 Completeness
2.2.1 Definition. Completeness.
An lcsE is calledsequentially completeif every Cauchy sequence converges.
It is calledcompletewhen every Cauchy net converges. A net (or sequence)xiis called Cauchyifxi´xjÑ0 for i, jÑ 8, i.e.
@εą0@pDi0@i, jąi0:ppxi´xjq ăε.
