Page 27 - Vector Analysis

§1.6 Bounded Linear Maps 23

Theorem 1.73. Let A be an nˆn matrix, and δ be an operator satisfying δ(f g) = f δg+(δf )g

whenever the product makes sense. Then

() (1.3)
δ det(A) = tr Adj(A)δA ,

where δA ” [δaij]nˆn if A = [aij]nˆn. In particular, if A is invertible,

δ det(A) = det(A)tr((A´1δA) .

[]
f (x) g(x) and δ = d . Then
Example 1.74. Let A(x) = h(x) k(x)
dx

( [ ][ ] ) ( [ ´ gh1 kg1 ´ gk1 ] )
tr k ´g f 1 g1 tr kf 1 1 + fh1 ´hg 1 + f k 1
δ det(A) = ´h f h1 k1 = ´hf

= kf 1 ´ gh1 ´ hg 1 + f k 1 .

1.6 Bounded Linear Maps

Definition 1.75 (Linear map). Let V and W be two vector spaces over a scalar field F. A
map L : V Ñ W is called a linear map from V into W if

L(αv + w) = αL(v) + L(w) @ α P F and v, w P V.

For notational convenience, we often write Lv instead of L(v). When V and W are finite
dimensional, linear maps (from V into W) are sometimes called linear transformations
(from V into W).

Let L1, L2 : V Ñ W be two linear maps, and α P F be a scalar. It is easy to see that
αL1 + L2 : V Ñ W is also a linear map. This is equivalent to say that the collection of
linear maps is a vector space, and this induces the following

Definition 1.76. The vector space L (V, W) is the collection of linear maps from V to W.

Definition 1.77 (Boundedness of linear maps). Let (V, }¨}V) and (W, }¨}W) be two normed

vector spaces over a scalar field F. A linear map L : V Ñ W is said to be bounded if the

number }Lv}W
}v}V
}L}B(V ,W ) ” sup }Lv}W = sup (1.4)

}v}V =1 v‰0

is finite. The collection of all bounded linear map from V to W is denoted by B(V, W),

and B(V, V) is also denoted by B(V) for simplicity.