Is a 2-dimensional subspace always called a plane no matter what the dimensions of the space is? Is a 2-dimensional subspace in a 7-dimensional space still called a plane? I know that a 6-dimensional space in 7-dimensional space is called a hyperplane because the difference in the number of dimensions of the space and subspace is 1. The answer should be easily googlable, but for some reason it's eluding me. Thanks!
 A: i guess this is all convention, but i feel safe to say:
Name of linear spaces (i.e. not curved):


*

*Dim=1: line

*Dim=2: plane

*Codim=1: hyperplane
When the space is not linear:


*

*Dim=1: curve

*Dim=2: surface

*Codim=1: hypersurface.
Codimenion is just a name for that difference in dimension you mentioned. So a hyperplane in a 2 dimensional space is in fact a line, even weirder a hyperplane in a 1 dimensional space is a point... When the hypersurface is given by a polynomial of degree $d$, it is common to refer to it as quadric ($d=2$), cubic ($d=3$), etc.
Edit: I claim no knowledge of terminology when the spaces are infinite dimensional.
A: From my understanding it is. It is analogous to points and lines, which also just convey a concept invariant of the dimensionality of the space they are embedded in. I think hyperplane is a more confusing term because it is not a plan and presumably is called hyperplane because it separates a n-dimensional space into to parts and thus the actual subspace it describes depends on the dimensionality of the space. 
But of course the two-dimensional sub-space has to be flat in order for it to be a plane.
