Page 29 - Vector Analysis

§1.6 Bounded Linear Maps 25

Definition 1.85 (Dual operator). Let V and W be two inner product spaces, and L : V Ñ
W be a bounded linear map. The dual operator of L, denoted by L˚, is the unique linear
map from W into V satisfying

(Lv, w)W = (v, L˚w)V @ v P V, w P W,

where (¨, ¨)V and (¨, ¨)W are inner products on V and W, respectively.

Definition 1.86 (Symmetry of linear maps). An linear map L P B(H) is said to be
symmetric if L = L˚ .

The last part of this section contributes to the following theorem which states that every
bounded linear maps near by (measured by the operator norm) an invertible bounded linear
map is also invertible.

Theorem 1.87. Let GL(n) be the set of all invertible linear maps on (Rn, } ¨ }2); that is,

GL(n) = ␣L P L (Rn, Rn) ˇ L is one-to-one (and onto)( .
ˇ

1. If L P GL(n) and K P B(Rn, Rn) satisfying }K ´ L}}L´1} ă 1 , then K P GL(n).

2. The mapping L ÞÑ L´1 is continuous on GL(n); that is,

@ ε ą 0 , D δ ą 0 Q }K´1 ´ L´1} ă ε whenever }K ´ L} ă δ .

Proof. 1. Let }L´1} = 1 and }K ´ L} = β. Then β ă α; thus for every x P Rn,

α

α}x}Rn = α}L´1Lx}Rn ď α}L´1}}Lx}Rn = }Lx}Rn ď }(L ´ K)x}Rn + }Kx}Rn
ď β}x}Rn + }Kx}Rn .

As a consequence, (α ´ β)}x}Rn ď }Kx}Rn and this implies that K : Rn Ñ Rn is
one-to-one hence invertible.

! 1 ε )
2}L´1} 2}L´1}2 .
2. Let L P GL(n) and ε ą 0 be given. Choose δ = min , If }K´L} ă δ,

then K P GL(n). Since L´1 ´ K´1 = K´1(K ´ L)L´1, we find that if }K ´ L} ă δ,

}K ´1 } ´ }L´1} ď }K ´1 ´ L´1} ď }K ´1 }}K ´ L}}L´1} ă 1 }K ´1}
2

which implies that }K´1} ă 2}L´1}. Therefore, if }K ´ L} ă δ,

}L´1 ´ K´1} ď }K´1}}K ´ L}}L´1} ă 2}L´1}2δ ă ε . ˝