Thursday, 10 January 2013

functional analysis - preserving norm map between real normed spaces



Suppose X,Y are two real normed spaces,T:XY is the bijective map such that ||Tx+Ty+Tz||=||x+y+z|| for any x,y,zX.Is T a linear map?


Answer



By assumption

so
T(x) + T(y) = -T(-x-y).
In particular, with x=y=0 it follows T(0)=0, and with y=0,
T(x) = -T(-x).
Hence
T(x) + T(y) = -T(-x-y) =T(x+y),

and T is additive.
Let me show that T is continous. Let x_n\to x.
Then
\|T(x) - T(x_n) \| = \|T(x) + T(-x_n)\|=\|x-x_n\|\to0.
So T is continuous and additive, hence it is linear, see Continuous and additive implies linear


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