Extending Continuous Sublinear maps on dense subsets of a Banach space

2011-04-18T22:47:32Z

Suppose X' and Y are Banach spaces and X is a linear subspace dense in X'. Let T be a continuous map of X to Y satisfying:

(1) ||T(x+y)|| is less than or equal to ||T(x)||+||T(y)||.

Please prove whether or not in general it is true that T has a continuous extension to X'.

Please also answer the same question for the case where we assume in addition that

(2) ||T(cx)||=|c|*||T(x)|| for all c real.

If so far, the answer has been "no," then assume that Y is an L^p space on some sigma finite measure space, with p>1. (Let me know if it's true for p=1 too.) If instead of using the p norm in the definitions (1) of subadditivity or (1) and (2) of sublinearity, we have pointwise almost everywhere inequalities of absolute values, then can T be extended?

To be completely explicit, in this case we would have

|T(x+y)| is less than or equal to |T(x)|+|T(y)| and |T(cx)|=|c|*|T(x)| for all c real pointwise almost everywhere on the measure space.

Answer by Ady for Extending Continuous Sublinear maps on dense subsets of a Banach space

2011-05-02T01:08:34Z

To 1) and 2). Let $X^{'}=Y=c_{0}$ , and let $X=c_{00}$. Take some $p\in X^{'}\smallsetminus X$, and define $T:X\rightarrow Y$ by $Tx:=\left\Vert x\right\Vert \cdot\left(\sin\left(\left\Vert x-p\right\Vert ^{-1}\right),1,0,0,0,...\right)$.

Then $T$ satisfies your conditions, yet it is not continuously extendable.

Extending Continuous Sublinear maps on dense subsets of a Banach space - MathOverflow

Extending Continuous Sublinear maps on dense subsets of a Banach space

Answer by Ady for Extending Continuous Sublinear maps on dense subsets of a Banach space