{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

2.6 Dual Spaces 3

# 2.6 Dual Spaces 3 - 2.6 Since T t f = f T and T(W W f T(W...

This preview shows pages 1–3. Sign up to view the full content.

2.6. Since T t f = fT and T ( W ) W, fT ( W ) f ( W ) = 0 T t f W T t ( W ) W W is T t -invariant ( ) If T ( W ) * W, w W s.t. T ( w ) W by the exercise 13, f W s.t. f ( T ( w )) 6 = 0 T t f ( w ) = fT ( w ) 6 = 0 i.e. f W s.t. T t f / W W is T -invariant 18. Φ : V * → L ( S, F ) (Actually L ( S, F ) ≡ L ( V, F )) (i) Clearly L ( S, F ) is a vector space over F and (ii) Φ is a linear map d f, g ∈ L ( S, F ) , Φ( f + g ) = ( f + g ) | s = f | s + g | s = Φ( f ) + Φ( g ) Φ( αf ) = αf | s = αf s = α Φ( f ) c (iii) f ker Φ , Φ( f ) = f s = 0 f S = ( spanS ) = V = { 0 } ker Φ = { 0 } 136 PNU-MATH

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2.6. Φ is one-to-one (iv) By the exercise 34 in section 2.1 ( f s : S F, ! f : V F a linear map s.t. f ( x ) = f s ( x ) , x S ) f s ∈ L ( S, F ) , ! f V * s.t. Φ( f ) = f s Φ is onto 19. (i) Choose y V, y / W and let γ : a basis of W then γ ∪ { y } : linearly independent by (section 1.7 or) Maximal principle, β : a basis of V s.t. γ ∪ { y } ⊆ β Define a function : g : β F s.t. g ( x ) = 0 x β, x 6 = y g ( y ) = 1 then by the exercise 18, f ∈ L ( V, F
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

2.6 Dual Spaces 3 - 2.6 Since T t f = f T and T(W W f T(W...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online