# How would I express the statement “Let H be a subspace of V” in mathematical notation?

How would I express the statement "Let H be a subspace of V" in mathematical notation? Does something like this work? $$( \ \ H(\mathbb{R})\subset V(\mathbb{R}) \ )$$

• I have seen my professor use $\leq$ in this context, You should probably include your own notation. Note that the subset sign somewhat drops the sub space property. – AlexR Apr 17 '14 at 23:48
• The notation $\le$ is common. If you use it, it may be a good idea to introduce it before you start mentioning it: "We will write $U\le V$ to indicate that $U$ is a vector subspace of the vector space $V$", or words to that effect. This should avoid any misunderstandings. Anyway, you should not use $\subset$ for this relation, as you may be interested in many subsets that are not vector spaces (bases, affine spaces, etc), and in that case the notation would just be ambiguous and not very useful. – Andrés E. Caicedo Apr 18 '14 at 0:24

The clearest way is to simply use "Let $H$ be a subspace of $V$". Then there is no need to use any specially defined notation.

• I've been studying the logic of proofs and math notation in preparation for an upper division linear algebra class. I know it may sound crazy but I really prefer the mathematical notation. The fact that "If P (is true), then Q (is true)." has the same meaning as "P is true only if Q is true." gets me nutty. Am I really crazy to think the former sounds like Q implies P. I'm just stupid, I'm sure I'll get used to it. It's so much clearer to just write $P\implies Q$. I know what I asked about is a case of a simple and clear statement but I just wanted to develop the form of mathematical notation. – vajra78 Apr 18 '14 at 6:08
• @vajra78: Please be careful about this tendency. I think it is perfectly natural for a student at the linear algebra level to want this sort of formalism. I certainly did, but when I tried to carry that wanting into analysis I was completely overwhelmed. You may be better than me and be able to handle analysis well, but eventually either you or your readers will simply not be able to handle this any longer. If by that point you are "addicted" to the notation then you will have to unlearn this tendency to proceed. This happened to a friend of mine: it is mathematically and emotionally hard. – Eric Stucky Aug 13 '14 at 5:03

In such a simple way, it suffices to write let $H$ be a subspace of $V$. In the context of "chains" of subspaces, you can introduce a terminology like

For vector spaces $U, V$ we write $$U\le V \quad :\Leftrightarrow U \text{ is a (closed) subspace of } V$$

This allows for more convenient notations like $$V_1 \le V_2 \le \ldots \le V_n\le H$$ For Galerkin-type algorithms (working on a chain of finite-dimensional closed subspaces)

• Thanks. I like that. Probably the programer in me. – vajra78 Apr 18 '14 at 6:26
• @vajra78 A numerical analysis professor introduced this notation to me, very close to a programmer ^^. Feel free to ask for further reference or accept the answer. – AlexR Apr 18 '14 at 6:27