Is euclidean space a linear space where inner product exists?
If $V$ is a complex linear space, then do there complex numbers $x,y$ in $V$ such that $\langle x,y \rangle$ is complex?
And
I don't understand this transformation.
$T: V\to E$, where $V$ is a subspace of Euclidean space.
Is $V$ a linear space? and why is it a subspace of Euclidean space?
To be honest,I don't completely understand your question. V is a subset of E where it's closed under addition and 0 is in V. A subspace by definition is also a linear or vector space. So T is a local isomorphism(linear transformation) of E from the subspace to the larger space.
On a complex vector space, we can certainly define a complex inner product in addition to the usual Euclidean inner product. The inner product can be defined completely abstractly via axioms so it's irrelevant whether the spaces are over real or complex fields.
For more details,take a look at this: