Functional Analysis Note 4 Functional / May 23, 2008

Functional Analysis

Note 4

Functional 

TANAKA Akio

1 Linear functional

Real number or complex number     Φ

Linear space over Φ     X

x∈X

f (x)∈Φ.    

(i) f (x₁+x₂) =f (x₁)+f (x₂)   (x₁, x₂∈X)

(ii) f (ax) =af (x)  (x∈X, a∈Φ)

2 Hyperspace

Linear space     X

Linear functional defined by X     f

N_f = {x∈X ; f (x) = 0 )}

x₀ ∉ N_f

Arbitrary x∈X

x = z + ax₀  ( z ∈N_f , a∈Φ )

Hypersurface is defined by the next.

N_f  + x₀ = { z + x₀ ;  z∈N_f }

3 Distance of hypersurface

Normed space     X

Linear functional defined at X     f

Bounded linear subspace     N_f  

N_f  ≠X

x₀ ∈X

f (x₀ )= 1

Hypersurface     M _f = N_f  + x₀ ( = { x∈X ; f (x) = 1 }

M _f ≠0

Distance between origin 0 and M _f     d

d = inf {||x|| ; x∈M _f } > 0

4 Bounded linear functional and distance

Bounded linear functional defined at normed space X     f

f (x₀ ) ≠ 0

M _f = { x∈X ; f (x) = 1 }

Distance between origin 0 and M _f     d

|| f || = 1/d

5 Closed linear subspace

Normed space     X

Closed linear subspace of X     M

Linear subspace of X and having finite dimension     V

M + V = { Z + v ; z∈M, v∈V }

Tokyo May 23, 2008

Sekinan Research Field of Language