Note 4
Functional
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
Nf = {x∈X ; f (x) = 0 )}
x0 ∉ Nf
Arbitrary x∈X
x = z + ax0 ( z ∈Nf , a∈Φ )
Hypersurface is defined by the next.
Nf + x0 = { z + x0 ; z∈Nf }
3 Distance of hypersurface
Normed space X
Linear functional defined at X f
Bounded linear subspace Nf
Nf ≠X
x0 ∈X
f (x0 )= 1
Hypersurface M f = Nf + 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
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