Saturday 28 April 2018

Functional Analysis Note 4 Functional


Note 4
Functional 


1 Linear functional
Real number or complex number     Φ
Linear space over Φ     X
xX
(x)Φ.    
(i) (x1+x2) =(x1)+f (x2)   (x1x2X)
(ii) f (ax) =af (x)  (xXaΦ)

2 Hyperspace
Linear space     X
Linear functional defined by X     f
N= {xX ; f (x) = 0 )}
x0  Nf
Arbitrary xX
x = z + ax0  ( z NaΦ )
Hypersurface is defined by the next.
N x0 = { z + x0 ;  zN}

3 Distance of hypersurface
Normed space     X
Linear functional defined at X     f
Bounded linear subspace     N 
N X
xX
(x0 )1
Hypersurface     M f N x( = { xX ; f (x) = 1 }
M f 0
Distance between origin 0 and M f     d
d = inf {||x|| ; xM f } > 0

4 Bounded linear functional and distance
Bounded linear functional defined at normed space     f
f (x 0
M f = { xX ; f (x) = 1 }
Distance between origin 0 and M f     d
|| || = 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 ; zM, v}

Tokyo May 23, 2008

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