14.5. EXERCISES 407

as follows. (w,∫ b

az(t)dt

)≡∫ b

a(w,z(t))dt.

Show that this definition is well defined and furthermore the triangle inequality,∣∣∣∣∫ b

az(t)dt

∣∣∣∣≤ ∫ b

a|z(t)|dt,

and fundamental theorem of calculus,

ddt

(∫ t

az(s)ds

)= z(t)

hold along with any other interesting properties of integrals which are true.

21. For V,W two inner product spaces, define∫ b

aΨ(t)dt ∈L (V,W )

as follows. (w,∫ b

aΨ(t)dt (v)

)≡∫ b

a(w,Ψ(t)v)dt.

Show this is well defined and does indeed give∫ b

a Ψ(t)dt ∈L (V,W ) . Also showthe triangle inequality ∣∣∣∣∣∣∣∣∫ b

aΨ(t)dt

∣∣∣∣∣∣∣∣≤ ∫ b

a||Ψ(t)||dt

where ||·|| is the operator norm and verify the fundamental theorem of calculus holds.(∫ t

aΨ(s)ds

)′= Ψ(t) .

Also verify the usual properties of integrals continue to hold such as the fact theintegral is linear and ∫ b

aΨ(t)dt +

∫ c

bΨ(t)dt =

∫ c

aΨ(t)dt

and similar things. Hint: On showing the triangle inequality, it will help if you usethe fact that

|w|W = sup|v|≤1|(w,v)| .

You should show this also.

22. Prove Gronwall’s inequality. Suppose u(t)≥ 0 and for all t ∈ [0,T ] ,

u(t)≤ u0 +∫ t

0Ku(s)ds.