What are some examples of vector spaces that aren't graded?

Question

From wikipedia: a vector space $V$ is graded if it decomposes into direct sum $ \oplus_{n \geq 0} V_n$ of vector spaces $V_n$.

So as far as I understand things, any vector space with a countable basis is graded: Let $V$ be a vector space over a field $k$ with basis $\{v_n\}_{n\in\mathbb{N}}$, then $V = \oplus_{n\geq 0} k\cdot v_n$. Then the only vector spaces that I can think of that aren't obviously graded are things like $C(X)$, the space of continuous functions on some manifold $X$

Is this correct? are there any more? or do I not understand something?

Thanks

Answer 1

Grading isn't a property of a vector space: it's extra structure attached to a vector space, in the same way that a multiplication is an extra structure you attach to a set to make it a group. So this is a little like asking "what are some examples of sets that aren't groups?" (As it turns out, every set can be equipped with a group structure, and this is equivalent to the axiom of choice. But this is missing the point.)

Every vector space admits a trivial grading in which $V_0 = V$ and $V_n = 0$ for all $n \ge 1$. But we often encounter vector spaces (such as the space of polynomials) with natural gradings, they are usually nontrivial, and taking advantage of this extra structure is useful in various ways.