Extending Continuous Sublinear maps on dense subsets of a Banach space

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'.

(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.

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Do you really want assume continuity of $T$ in your second sentence? –  Yemon Choi Apr 18 '11 at 23:52

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