SOLUTIONS MANUAL FOR
Applied Functional
Analysis
THIRD EDITION
by
J. Tinsley Oden
Leszek F. Demkowicz
, 4
1
Preliminaries
Elementary Logic and Set Theory
1.1 Sets and Preliminary Notations, Number Sets
Exercises
Exercise 1.1.1 I f IZ = {. . . , −2, −1, 0, 1, 2,.. .} denotes the set of all integers and IN = {1, 2, 3,. . .} the set
of all natural numbers, exhibit the following sets in the form A = {a, b, c, . . .}:
(i) {x ∈ IZ : x − 2x + 1 = 0}
2
(ii) {x ∈ IZ : 4 ≤ x ≤ 10}
(iii) {x ∈ IN : x < 10}
2
(i) {1}
(ii) {4, 5, 6, 7, 8, 9, 10}
(iii) {1, 2, 3}
Itservesasacrucialfoundationforfurtherexplorationintocellularstructureandfunction,genetics,andphysiologicalmechanismscoveredinsubsequentchaptersofthetextbook.Ifyouhavespecificquestionsaboutanyo
fthesetopicsorwouldlikemoredetailedinformationonaparticularaspectofChapter3,feelfreetoask!Chapter4:InsidetheCellChapter4of"EssentialsofBiology"bySylviaMaderandMichaelWindelspecht,titled"Insi
detheCell,"typicallyfocusesonthestructureandfunctionsofcells,whicharethebasicunitsoflife.Here‘sanoverviewofwhatyoumightfindinthischapter:**IntroductiontoCells**:Definitionofcellsasthefundamental
unitsoflife,theirdiversityinstructureandfunction,andtheCellTheory.**CellStructure**:Explorationofthestructureoftypicaleukaryoticcells,including:**CellMembrane**:Structure,composition(lipidbilayer,
proteins),functionsincelltransport,signaling,andrecognition.**Cytoplasm**:Composition,organellessuspendedwithin(e.g.,cytoskeleton,ribosomes).**Nucleus**:Structure,functions(DNAstorage,transcrip
tion,regulationofcellactivities).**Organelles**:Detailedexaminationoforganellessuchasmitochondria,endoplasmicreticulum,Golgiapparatus,lysosomes,andtheirrolesincellularprocesses.**ProkaryoticCell
s**:Comparisonofprokaryoticandeukaryoticcellstructures,emphasizingdifferencesinorganelles,geneticmaterial(nonucleusinprokaryotes),andcellularprocesses.**CellularOrganizationandFunction**:Integ
rationofcellularcomponentsandorganellestosupportcellularfunctionssuchasmetabolism,proteinsynthesis,energyproduction(e.g.,ATPsynthesis),andmaintenanceofhomeostasis.**CellularTransport**:Mech
anismsofcellulartransport,including:**PassiveTransport**:Diffusion,osmosis,facilitateddiffusion.**ActiveTransport**:Ionpumps,carrierproteins,endocytosis,exocytosis.**BulkTransport**:Phagocytosi
s,pinocytosis.**CellCommunication**:Overviewofcellularcommunicationprocesses,including:**ReceptorProteins**:Recognitionofsignalingmolecules(ligands).**SignalTransduction**:Transmissionof
signalswithincells(e.g.,secondmessengers).**CellSignalingPathways**:Examplesofsignalingpathways(e.g.,hormonesignaling,neurotransmission).**CellCycleandCellDivision**:Phasesofthecellcycle(int
erphase,mitosis,cytokinesis)andtheirregulation.Importanceofcelldivisioningrowth,repair,andreproduction.**CancerandCellRegulat ion**:Introductiontocancerasuncontrolledcellgrowthanddivision.Cause
sofcancer(mutations,environmentalfactors),mechanismsoftumorformation,andapproachestocancertreatment.**StemCellsandRegenerativeMedicine**:Overviewofstemcells,theirproperties(pluripotent,mu
ltipotent),andpotentialapplicationsinregenerativemedicineandresearch.**MicroscopyandCellVisualization**:Techniquesandtypesofmicroscopyusedtostudycells,includinglightmicroscopy,electronmicros
copy
, Preliminaries 5
1.2 Level One Logic
Exercises
Exercise 1.2.1 Construct the truth table for De Morgan’s Law:
~ (p ∧ q) ⇔ ((∼ p) ∨ (∼ q))
1
(p ∧ q) ⇔ (( ∼ p) ∨ (∼ q))
1 000 1 10 1 10
1 001 1 10 1 01
1 100 1 01 1 10
0 111 1 01 0 01
Exercise 1.2.2 Construct truth tables to prove the following tautologies:
(p ⇒ q) ⇔ (∼ q ⇒∼ p)
∼ (p ⇒ q) ⇔ p ∧ ∼ q
(p ⇒ q) ⇔ ( ∼ q ⇒∼ p)
0 1 0 1 10 1 10
0 1 1 1 01 1 10
1 0 0 1 10 0 01
1 1 1 1 01 1 01
(p q) p q
0 (0 1 0) 1 0 0 1 0
0 (0 1 1) 1 0 0 0 1
1 (1 0 0) 1 1 1 1 0
0 (1 1 1) 1 1 0 0 1
Exercise 1.2.3 Construct truth tables to prove the associative laws in logic:
p ∨ (q ∨ r) ⇔ (p ∨ q) ∨ r
p ∧ (q ∧ r) ⇔ (p ∧ q) ∧ r
p (q r) (p q) r
00 000 1 000 00
01 011 1 000 11
01 110 1 011 10
01 111 1 011 11
11 000 1 110 10
11 011 1 110 11
11 110 1 111 10
11 111 1 111 11
p (q r) (p q) r
, 6
0 0 000 1 000 00
0 0 001 1 000 01
0 0 100 1 001 00
0 0 111 1 001 01
1 0 000 1 100 00
1 0 001 1 100 01
1 0 100 1 111 00
1 1 111 1 111 11
1.3 Algebra of Sets
Exercises
Exercise 1.3.1 Of 100 students polled at a certain university, 40 were enrolled in an engineering course,
50 in a mathematics course, and 64 in a physics course. Of these, only 3 were enrolled in all three
subjects, 10 were enrolled only in mathematics and engineering, 35 were enrolled only in physics and
mathematics, and 18 were enrolled only in engineering and physics.
Itservesasacrucialfoundationforfurtherexplorationintocellularstructureandfunction,genetics,andphysiologicalmechanismscoveredinsubsequentchaptersofthetextbook.Ifyouhavespeci
ficquestionsaboutanyofthesetopicsorwouldlikemoredetailedinformationonaparticularaspectofChapter3,feelfreetoask!Chapter4:InsidetheCellChapter4of"EssentialsofBiolo
gy"bySylviaMaderandMichaelWindelspecht,titled"InsidetheCell,"typicallyfocusesonthestructureandfunctionsofcells,whicharethebasicunitsoflife.Here‘sanoverviewofwh
atyoumightfindinthischapter:**IntroductiontoCells**:Definitionofcellsasthefundamentalunitsoflife,theirdiversityinstructureandfunction,andtheCellTheory.**CellStructur
e**:Explorationofthestructureoftypicaleukaryoticcells,including:**CellMembrane**:Structure,composition(lipidbilayer,proteins),functionsincelltransport,signaling,andr
ecognition.**Cytoplasm**:Composition,organellessuspendedwithin(e.g.,cytoskeleton,ribosomes).**Nucleus**:Structure,functions(DNAstorage,transcription,regulation
ofcellactivities).**Organelles**:Detailedexaminationoforganellessuchasmitochondria,endoplasmicreticulum,Golgiapparatus,lysosomes,andtheirrolesincellularprocesses.
**ProkaryoticCells**:Comparisonofprokaryoticandeukaryoticcellstructures,emphasizingdifferencesinorganelles,geneticmaterial(nonucleusinprokaryotes),andcellularpro
cesses.**CellularOrganizationandFunction**:Integrationofcellularcomponentsandorganellestosupportcellularfunctionssuchasmetabolism,proteinsynthesis,energyproduct
ion(e.g.,ATPsynthesis),andmaintenanceofhomeostasis.**CellularTransport**:Mechanismsofcellulartransport,including:**PassiveTransport**:Diffusion,osmosis,facilitat
eddiffusion.**ActiveTransport**:Ionpumps,carrierproteins,endocytosis,exocytosis.**BulkTransport**:Phagocytosis,pinocytosis.**CellCommunication**:Overviewofc
ellularcommunicationprocesses,including:**ReceptorProteins**:Recognitionofsignalingmolecules(ligands).**SignalTransduction**:Transmissionofsignalswithincells(e.
g.,secondmessengers).**CellSignalingPathways**:Examplesofsignalingpathways(e.g.,hormonesignaling,neurotransmission).**CellCycleandCellDivision**:Phasesofthe
cellcycle(interphase,mitosis,cytokinesis)andtheirregulation.Importanceofcelldivisioningrowth,repair,andreproduction.**CancerandCellRegulation**:Introductiontocance
rasuncontrolledcellgrowthanddivision.Causesofcancer(mutations,environmentalfactors),mechanismsoftumorformation,andapproachestocancertreatment.**StemCellsand
RegenerativeMedicine**:Overviewofstemcells,theirproperties(pluripotent,multipotent),andpotentialapplicationsinregenerativemedicineandresearch.**MicroscopyandCel
lVisualization**:Techniquesandtypesofmicroscopyusedtostudycells,includinglightmicroscopy,electronmicroscopy
(i) How many students were enrolled only in mathematics?
(ii) How many of the students were not enrolled in any of these three subjects?
Let A, B, C denote the subsets of students enrolled in mathematics, the engineering course and physics,
repectively. Sets: A ∩ B ∩ C, A ∩ B − (A ∩ B ∩ C),A ∩ C − (A ∩ B ∩ C) and A − (B ∪ C) are
pairwise disjoint (no two sets have a nonempty common part) and their union equals set A, see Fig. 1.1.
Consequently,
#(A − (B ∪ C)) = #A − #A ∩ B ∩ C − #(A ∩ B − (A ∩ B ∩ C)) − #(A ∩ C − (A ∩ B ∩ C))
= 50 − 3 − 10 − 35 = 2
In the same way we compute,
#(B − (A ∪ C)) = 9 and #(C − (A ∪ B)) = 8
Applied Functional
Analysis
THIRD EDITION
by
J. Tinsley Oden
Leszek F. Demkowicz
, 4
1
Preliminaries
Elementary Logic and Set Theory
1.1 Sets and Preliminary Notations, Number Sets
Exercises
Exercise 1.1.1 I f IZ = {. . . , −2, −1, 0, 1, 2,.. .} denotes the set of all integers and IN = {1, 2, 3,. . .} the set
of all natural numbers, exhibit the following sets in the form A = {a, b, c, . . .}:
(i) {x ∈ IZ : x − 2x + 1 = 0}
2
(ii) {x ∈ IZ : 4 ≤ x ≤ 10}
(iii) {x ∈ IN : x < 10}
2
(i) {1}
(ii) {4, 5, 6, 7, 8, 9, 10}
(iii) {1, 2, 3}
Itservesasacrucialfoundationforfurtherexplorationintocellularstructureandfunction,genetics,andphysiologicalmechanismscoveredinsubsequentchaptersofthetextbook.Ifyouhavespecificquestionsaboutanyo
fthesetopicsorwouldlikemoredetailedinformationonaparticularaspectofChapter3,feelfreetoask!Chapter4:InsidetheCellChapter4of"EssentialsofBiology"bySylviaMaderandMichaelWindelspecht,titled"Insi
detheCell,"typicallyfocusesonthestructureandfunctionsofcells,whicharethebasicunitsoflife.Here‘sanoverviewofwhatyoumightfindinthischapter:**IntroductiontoCells**:Definitionofcellsasthefundamental
unitsoflife,theirdiversityinstructureandfunction,andtheCellTheory.**CellStructure**:Explorationofthestructureoftypicaleukaryoticcells,including:**CellMembrane**:Structure,composition(lipidbilayer,
proteins),functionsincelltransport,signaling,andrecognition.**Cytoplasm**:Composition,organellessuspendedwithin(e.g.,cytoskeleton,ribosomes).**Nucleus**:Structure,functions(DNAstorage,transcrip
tion,regulationofcellactivities).**Organelles**:Detailedexaminationoforganellessuchasmitochondria,endoplasmicreticulum,Golgiapparatus,lysosomes,andtheirrolesincellularprocesses.**ProkaryoticCell
s**:Comparisonofprokaryoticandeukaryoticcellstructures,emphasizingdifferencesinorganelles,geneticmaterial(nonucleusinprokaryotes),andcellularprocesses.**CellularOrganizationandFunction**:Integ
rationofcellularcomponentsandorganellestosupportcellularfunctionssuchasmetabolism,proteinsynthesis,energyproduction(e.g.,ATPsynthesis),andmaintenanceofhomeostasis.**CellularTransport**:Mech
anismsofcellulartransport,including:**PassiveTransport**:Diffusion,osmosis,facilitateddiffusion.**ActiveTransport**:Ionpumps,carrierproteins,endocytosis,exocytosis.**BulkTransport**:Phagocytosi
s,pinocytosis.**CellCommunication**:Overviewofcellularcommunicationprocesses,including:**ReceptorProteins**:Recognitionofsignalingmolecules(ligands).**SignalTransduction**:Transmissionof
signalswithincells(e.g.,secondmessengers).**CellSignalingPathways**:Examplesofsignalingpathways(e.g.,hormonesignaling,neurotransmission).**CellCycleandCellDivision**:Phasesofthecellcycle(int
erphase,mitosis,cytokinesis)andtheirregulation.Importanceofcelldivisioningrowth,repair,andreproduction.**CancerandCellRegulat ion**:Introductiontocancerasuncontrolledcellgrowthanddivision.Cause
sofcancer(mutations,environmentalfactors),mechanismsoftumorformation,andapproachestocancertreatment.**StemCellsandRegenerativeMedicine**:Overviewofstemcells,theirproperties(pluripotent,mu
ltipotent),andpotentialapplicationsinregenerativemedicineandresearch.**MicroscopyandCellVisualization**:Techniquesandtypesofmicroscopyusedtostudycells,includinglightmicroscopy,electronmicros
copy
, Preliminaries 5
1.2 Level One Logic
Exercises
Exercise 1.2.1 Construct the truth table for De Morgan’s Law:
~ (p ∧ q) ⇔ ((∼ p) ∨ (∼ q))
1
(p ∧ q) ⇔ (( ∼ p) ∨ (∼ q))
1 000 1 10 1 10
1 001 1 10 1 01
1 100 1 01 1 10
0 111 1 01 0 01
Exercise 1.2.2 Construct truth tables to prove the following tautologies:
(p ⇒ q) ⇔ (∼ q ⇒∼ p)
∼ (p ⇒ q) ⇔ p ∧ ∼ q
(p ⇒ q) ⇔ ( ∼ q ⇒∼ p)
0 1 0 1 10 1 10
0 1 1 1 01 1 10
1 0 0 1 10 0 01
1 1 1 1 01 1 01
(p q) p q
0 (0 1 0) 1 0 0 1 0
0 (0 1 1) 1 0 0 0 1
1 (1 0 0) 1 1 1 1 0
0 (1 1 1) 1 1 0 0 1
Exercise 1.2.3 Construct truth tables to prove the associative laws in logic:
p ∨ (q ∨ r) ⇔ (p ∨ q) ∨ r
p ∧ (q ∧ r) ⇔ (p ∧ q) ∧ r
p (q r) (p q) r
00 000 1 000 00
01 011 1 000 11
01 110 1 011 10
01 111 1 011 11
11 000 1 110 10
11 011 1 110 11
11 110 1 111 10
11 111 1 111 11
p (q r) (p q) r
, 6
0 0 000 1 000 00
0 0 001 1 000 01
0 0 100 1 001 00
0 0 111 1 001 01
1 0 000 1 100 00
1 0 001 1 100 01
1 0 100 1 111 00
1 1 111 1 111 11
1.3 Algebra of Sets
Exercises
Exercise 1.3.1 Of 100 students polled at a certain university, 40 were enrolled in an engineering course,
50 in a mathematics course, and 64 in a physics course. Of these, only 3 were enrolled in all three
subjects, 10 were enrolled only in mathematics and engineering, 35 were enrolled only in physics and
mathematics, and 18 were enrolled only in engineering and physics.
Itservesasacrucialfoundationforfurtherexplorationintocellularstructureandfunction,genetics,andphysiologicalmechanismscoveredinsubsequentchaptersofthetextbook.Ifyouhavespeci
ficquestionsaboutanyofthesetopicsorwouldlikemoredetailedinformationonaparticularaspectofChapter3,feelfreetoask!Chapter4:InsidetheCellChapter4of"EssentialsofBiolo
gy"bySylviaMaderandMichaelWindelspecht,titled"InsidetheCell,"typicallyfocusesonthestructureandfunctionsofcells,whicharethebasicunitsoflife.Here‘sanoverviewofwh
atyoumightfindinthischapter:**IntroductiontoCells**:Definitionofcellsasthefundamentalunitsoflife,theirdiversityinstructureandfunction,andtheCellTheory.**CellStructur
e**:Explorationofthestructureoftypicaleukaryoticcells,including:**CellMembrane**:Structure,composition(lipidbilayer,proteins),functionsincelltransport,signaling,andr
ecognition.**Cytoplasm**:Composition,organellessuspendedwithin(e.g.,cytoskeleton,ribosomes).**Nucleus**:Structure,functions(DNAstorage,transcription,regulation
ofcellactivities).**Organelles**:Detailedexaminationoforganellessuchasmitochondria,endoplasmicreticulum,Golgiapparatus,lysosomes,andtheirrolesincellularprocesses.
**ProkaryoticCells**:Comparisonofprokaryoticandeukaryoticcellstructures,emphasizingdifferencesinorganelles,geneticmaterial(nonucleusinprokaryotes),andcellularpro
cesses.**CellularOrganizationandFunction**:Integrationofcellularcomponentsandorganellestosupportcellularfunctionssuchasmetabolism,proteinsynthesis,energyproduct
ion(e.g.,ATPsynthesis),andmaintenanceofhomeostasis.**CellularTransport**:Mechanismsofcellulartransport,including:**PassiveTransport**:Diffusion,osmosis,facilitat
eddiffusion.**ActiveTransport**:Ionpumps,carrierproteins,endocytosis,exocytosis.**BulkTransport**:Phagocytosis,pinocytosis.**CellCommunication**:Overviewofc
ellularcommunicationprocesses,including:**ReceptorProteins**:Recognitionofsignalingmolecules(ligands).**SignalTransduction**:Transmissionofsignalswithincells(e.
g.,secondmessengers).**CellSignalingPathways**:Examplesofsignalingpathways(e.g.,hormonesignaling,neurotransmission).**CellCycleandCellDivision**:Phasesofthe
cellcycle(interphase,mitosis,cytokinesis)andtheirregulation.Importanceofcelldivisioningrowth,repair,andreproduction.**CancerandCellRegulation**:Introductiontocance
rasuncontrolledcellgrowthanddivision.Causesofcancer(mutations,environmentalfactors),mechanismsoftumorformation,andapproachestocancertreatment.**StemCellsand
RegenerativeMedicine**:Overviewofstemcells,theirproperties(pluripotent,multipotent),andpotentialapplicationsinregenerativemedicineandresearch.**MicroscopyandCel
lVisualization**:Techniquesandtypesofmicroscopyusedtostudycells,includinglightmicroscopy,electronmicroscopy
(i) How many students were enrolled only in mathematics?
(ii) How many of the students were not enrolled in any of these three subjects?
Let A, B, C denote the subsets of students enrolled in mathematics, the engineering course and physics,
repectively. Sets: A ∩ B ∩ C, A ∩ B − (A ∩ B ∩ C),A ∩ C − (A ∩ B ∩ C) and A − (B ∪ C) are
pairwise disjoint (no two sets have a nonempty common part) and their union equals set A, see Fig. 1.1.
Consequently,
#(A − (B ∪ C)) = #A − #A ∩ B ∩ C − #(A ∩ B − (A ∩ B ∩ C)) − #(A ∩ C − (A ∩ B ∩ C))
= 50 − 3 − 10 − 35 = 2
In the same way we compute,
#(B − (A ∪ C)) = 9 and #(C − (A ∪ B)) = 8
