Does anyone know any nice examples of vector spaces without a basis that is in some sense "natural".
To clarify what I mean, suppose we look at $\mathbb{R}^2$. We define $\mathbb{R}^2$ as pairs of real numbers. In some sense, what we are doing is expressing vectors in terms of a natural basis : (1,0) and (0,1). This is not what I want.
An example that I thought of is a tangent space to a manifold. When one picks a tangent space to a manifold, there is no natural basis that one can pick.
Are there other nice examples?
Best Answer
Most vector spaces I've met don't have a natural basis. However this is question that comes up when teaching linear algebra. You want to motivate abstract vector spaces instead of working with $\mathbb{R}^n$ (or your favourite field in place of $\mathbb{R}$). One simple example, is this.
Consider $\mathbb{R}^n$ ($n>2$) as a euclidean space relative to the "dot" product and let $v = (1,1,\dots,1)$. Then the subspace $V \subset \mathbb{R}^n$ of vectors orthogonal to $v$ does not have a natural basis. If you don't like introducing an inner product, then take $V$ to be the annihilator of $v$ in the dual of $\mathbb{R}^n$. This actually comes up when discussing the root space of $\mathfrak{su}(n)$, say.