UPMC Basic functional analysis
Master 1, MM05E 2011-2012
Continuous linear maps
1) Let(E,k · kE)be a normed linear space of dimensionN. Show that any weakly converging sequence inE is also strongly converging.
2) Let(X,k · k)be a real normed linear space
a) Show that for all x∈X, there existsf ∈X0 such thatf(x) =kxk andkfkX0 = 1.Hint : Use the Hahn-Banach Theorem.
b) Show that for allx∈X,
kxk= max
f∈X0,kfk≤1f(x) = max
f∈X0,f6=0
f(x) kfkX0
.
c) Deduce thatX is isometrically isomorphic to a subspace of its bidualX00:= (X0)0 (i.e.there exists a subspaceXe ⊂X00and a one to one linear continuous mapJ :X →Xe such thatkJ(x)kX00=kxk for allx∈X). When isXe closed ?
Remark : IfXe =X00, we say thatX isreflexible. In particular, uniformly convex Banach spaces or Hilbert spaces are reflexible.
3) Let(X,k · k)be a normed linear space.
a) show that ifxn→x, thenxn * xandkxnk → kxk.
b) Show that if xn* x, then
kxk ≤lim inf
n→∞ kxnk
and that there exists a constantC >0 such thatkxnk ≤C for alln∈N. c) Show that if X is uniformly convex, then
xn * xandkxnk → kxk ⇒xn →x.
4) Let(X,k · k) be a uniformly convex Banach space, and letA⊂X be non empty closed and convex set.
a) Show that for allx∈X, there exists a uniqueax∈Asuch that kx−axk= dist(x, A) := inf
a∈Akx−ak.
b) Show that the map x7→ax is continuous.
5) Open mapping Theorem. Let E and F be two Banach spaces, and `∈ Lc(E, F) be a surjective continuous map. Then `is an open mapping, i.e., for every open set U ⊂E, then`(U)is open inF.
a) We denote by BE andBF the open unit balls inE andF respectively. Show that`(BE)has non empty interior inF (we can considerXn :=n `(BE)).
b) Deduce that there existsr >0, such that2rBF ⊂`(BE).
c) Lety∈rBF, show that there existsx1∈12BEsuch thaty1:=y−`(x1)∈ r2BF. Construct then two sequences(xn),(yn)such thatxn∈2−nBE and yn=yn−1−`(xn)∈2−nrBF. Deduce thaty∈`(2BE).
d) Show that
∃r >0, ∀y∈F,kykF < r,∃x∈E,kxk<2 and`(x) =y. (?) Deduce that for each open setU ⊂E, then`(U)is open inF.
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6) Banach Theorem. Let E and F be two Banach spaces and ` ∈ Lc(E, F) be a linear one to one continuous map. Show that`−1∈ Lc(F, E), and that k`−1kLc(F,E)≥1/k`kLc(E,F).
7) Closed graph Theorem. Let E andF be two Banach spaces, and T be a linear map from E to F. We suppose that the graph of T, G(T) :={(x, T x) :x∈E} is a closed subset of E×F. Then T is continuous.
8) LetF be a closed linear subspace ofC([0,1])which is contained inC1([0,1]).
a) Show that the derivation mapD:f ∈F →f0 ∈ C([0,1]) is continous.
b) Deduce thatF has finite dimension.
9) Grothendieck Theorem. Let(X,M)be a measure space andµbe a probability measure onM. Let S be a closed subspace of Lp(X, µ)(p >0), contained inL∞(X, µ). ThenS has finite dimension.
a) Show that there exists a constantK <∞such that for allf ∈S, thenkfk∞≤Kkfkp. b) Deduce that there exists a constantM <∞such that for allf ∈S, then kfk∞≤Mkfk2.
c) Show that forc:= (c1, . . . , cn)∈IRn withkck2≤1and forφ1, . . . , φn orthonormal inS, then the functionfc:=Pn
i=1ciφi satisfieskfck∞≤M.
d) Deduce that there exists X0 ⊂ X with µ(X0) = 1such that for all c := (c1, . . . , cn)∈ IRn with kck2≤1 and for allx∈X0, one has|fc(x)| ≤M.
e) Conclude.
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