The discussion revolves around the norm of a linear transformation, specifically the equivalence of two definitions of the norm: one involving the maximum of |T(x)| for |x| ≤ 1 and the other for |x| = 1. Participants are exploring the properties of bounded linear transformations in the context of this equivalence.
Some participants have provided insights into the properties of bounded linear transformations and the implications of linearity. There is an ongoing exploration of the relationship between the vectors involved, with no explicit consensus reached yet.
There is uncertainty regarding whether T is a bounded linear transformation, and some participants express this ambiguity. The discussion also touches on the implications of the definitions provided and the assumptions surrounding them.
radou said:I assume you are speaking of a bounded linear transformation?
If T is bounded, then there exists some constant C so that ||Tx||<=C||x|| for all x from the domain of T, and it follows almost directly from this definition.
CarmineCortez said:Tx >= T(x/||x||)
ystael said:[tex]Tx[/tex] and [tex]T\left(\frac{x}{\|x\|}\right)[/tex] are vectors in the range space of [tex]T[/tex]; they do not possesses an order relation. The relationship between these two vectors is simpler, and follows directly from the definition of a linear transformation.