# 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!

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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.

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Except for example, in the space of all continuous real valued functions on $\mathbb R$, it would be weird to call the subspace spanned by $e^x$ a line.. –  Ravi Donepudi Aug 6 '12 at 7:39
Well for infinite dimensional spaces i don't know the convention. Thanks, i added an edit. But for me, it would make sense to call the $\mathbb{R}$-span of $e^x$ a line; you need one parameter to specify a point on it. –  Joachim Aug 6 '12 at 8:05
@FortuonPaendrag, do you have the same problem with the linear span of a monomial? Polynomial vector spaces of bounded degree are finite-dimensional. –  alancalvitti Aug 12 '12 at 15:26
@alancalvitti : I am unsure, but I would just refer to it as the "subspace spanned by ____" –  Ravi Donepudi Aug 12 '12 at 18:33