Hello,
I was wondering if is a subspace of ? Some classmates of me tell me it is, some say it isn't.
I'm under the impression that it is in fact a subspace, anyone care to elaborate?
Thanks in advance
Sure it is. is a vector space in its own right, and it is a subset of Ergo, it's a subspace. The only confusion might be in a mix of notations. But you can just say, for example, that the third component is always zero (sort of an embedding scheme, I guess).
Does that help?
A necessary condition to be subspace of is that . Taking into account that we can't say strictly that is a subspace of .
However (and following Ackbeet's outline) you can define:
This function is an monomorphism (with usual operations) , so we can identify algebraically with (which is a subspace of . In that sense we say " is a subspace of " .
Fernando Revilla
Just to clarify what Ackbeet is saying:
Technically it is not a subspace, but it is isomorphic to one in a very strong sense (by adding an extra component which is always zero).
Most mathematicians will say it is a subspace by thinking in terms of this isomorphism (they are technicaly wrong by saying this, but this convention has been agreed upon in the mathematical community).