Q.4

Question 1

INTERIOR OF SET M

Answer 1

int(M) = union of open sets in M

Question 2

NOWHERE DENSE SET M

Answer 2

if int(M(-)) = ∅

Question 3

MEAGER SET

Answer 3

M that is union of nowhere dense sets Mi

Question 4

OPEN MAPPING THEOREM

Answer 4

if X and Y are banach spaces, then T∈B(X,Y) surjective ⟹ T is an open map
i.e. O⊆X open ⟹ T(O)⊆Y open

Question 5

BOUNDED INVERSE THEOREM

Answer 5

if X,Y banach and T∈B(X,Y)
then
T bijective ⟹ T^(-1)∈B(Y,X)

Question 6

CLOSED RANGE THEOREM

Answer 6

assume X,Y banach and T∈B(X,Y)
then
∃c>0: ||Tx||>=c||x||, ∀x∈X
⟺
T injective and ran T closed

Question 7

GRAPH OF T∈L(V,Y)

Answer 7

G(T) = {(x,Tx) : x∈V} ⊆ XxY

where X,Y NLS and V⊆X linear subspace

Question 8

CLOSED OPERATOR T

Answer 8

if G(T) closed in XxY

where X,Y NLS and V⊆X linear subspace

Question 9

P:X->X IS A PROJECTION ⟺

Answer 9

⟺ I-P is a projection
* in this case ranP = ker(I-P)
and kerP = ran(I-P)
furthermore X=ranP+kerP is a direct sum (i.e. ranP ∩ kerP = {0})