Vector spaces without natural bases

Question

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?

Answer 1

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.

Answer 2

The obvious example is $\mathbb{R}$, as a vector space over $\mathbb{Q}$; the existence of such a basis requires the axiom of choice.

Answer 3

The vector space $\mathbb C / \mathbb R$ does not have a preferred basis. Among the two bases $\{1, i\}$ and $\{1, -i\}$, there is no reason to prefer one over the other. The choice of one of these amounts to a choice of an orientation for the plane.

Answer 4

Another example is most function spaces defined over $\mathbb{R}$. The space of square integrable functions $L^2(\mathbb{R})$ doesn't have a natural basis. You would like one in the trigonometric functions $e^{2\pi i n x}$ in view of Plancherel's theorem and the Fourier transform, but they are not actually in $L^2$. (Compare the case on a torus, where the "natural" basis exists.)

Answer 5

The solution space of a homogeneous (ordinary or partial) linear differerential equation has no natural basis.

Answer 6

For teaching purposes, the most simple example (which I use frequently in a first course in linear algebra) is a generic sub-vector space of $\mathbb{R}^n$. Any vector plane in the $3$-space that is not cardinal works.

Answer 7

One example I like to use is the $1$-dimensional vector space of multiples of some physical unit (length, time, mass): for example, the meter is a basis of the $1$-dimensional vector space of lengths, and the light-year is also a basis of it, but there is no natural basis of this vector space.

This example can also be used to illustrate multilinear algebra constructs on $1$-dimensional vector spaces: the space of speeds is the (still $1$-dimensional) space of linear maps between the $1$-dimensional vector space of time spans and the $1$-dimensional vector space of lengths (it turns out that, in special relativity, but not in classical mechanics, there is a canonical isomorphism between these spaces, i.e., a canonical basis for the space of speeds). The space of areas it the tensor square of the space of lengths, and the space of volumes is its tensor cube. And so on.

This kind of example makes it clear why for $1$-dimensional vector spaces $V$, the tensor product of $V$ with its dual is canonically isomorphic to the base field, so such spaces can be called "invertible" (as in "invertible sheaf").

Answer 8

Hilbert spaces don't generally have nice bases in the sense of linear algebra. Neither does the ring of formal power series $k[[X]]$ over a field $k$. (These have "bases" with "infinite linear combinations" that only make sense because of completeness.)

Answer 9

The vector space of polynomials (possibly of some fixed degree). This is a case where many students, I think, are tempted to privilege the basis $\{ 1, x, x^2, ... \}$, but to do so is to 1) privilege evaluation at $0$ over evaluation at other points, and 2) miss out on the utility of other bases like $\{ 1, x, {x \choose 2}, ... \}$.

Answer 10

To expand on Anon's answer, I'd like to discuss one way in which the lack of a "natural" basis has some utility. A Hamel basis is a basis for $\mathbb{R}$ over $\mathbb{Q}$. Hamel bases are quite useful, due to their interactions with Cauchy functions (real-valued functions that satisfy an "additive" functional equation $f(x+y) = f(x) + f(y)$. This functional equation is equivalent to being linear over $\mathbb{Q}$. Examples of the utility of Cauchy functions abound. One approach to proving that the cube and the tetrahedron are not equidecomposable (Hilbert's 3rd problem) is to pick the $\mathbb{Q}-$linearly independent set $\{1, \pi\}$ and, by the magic of AC, this extends to a Hamel basis. Setting up the right Cauchy function then resolves the problem. For more on this, see "Conjecture and Proof" by Miklós Laczkovich.

Answer 11

I suppose there's a natural way to give a type of global quantative answer to this question. A vector bundle is a family of vector spaces over a base space, $f : E \to B$. $f$ is a continuous function, $B$ is a topological space and $f^{-1}(b)$ is a vector space for all $b\in B$. Moreover it is a continuous family of vector spaces in the sense that vector addition $E \oplus E \to E$ and scalar multiplication $\mathbb R \times E \to E$ are continuous.

If vector spaces typically had natural basis, vector bundles would typically be trivial. i.e. $E \simeq V \times B$ and under that homeomorphism, $f$ would be conjugate to projection $\pi : V \times B \to B$, $\pi(v,b) = b$, since choosing such a conjugation is equivalent to choosing (continuously) a basis for each vector space $f^{-1}(b)$. But this generally can't be done. The Moebius band being the first interesting counter-example. The non-triviality of the Moebius band from this perspective would be a reflection of the difficulty choosing a basis for 1-dimensional vector spaces.

Answer 12

Probably the simplest example --- $\{\ (x_1,...,x_n)\ |\ x_1+...+x_n=0\ \}$. Any choice of a basis distorts some symmetry...

As pointed out by Joe Silverman, this was in fact already mentioned in one of the previous answers. Let me still leave it, I think in this form it is sort of a "bare bones" example...

Answer 13

As a physicist, I would say the most obvious example is $n$-dimensional Euclidean space, with $n > 1$. Since a few people have mentioned casually that Euclidean spaces do have natural bases, I should explain myself...

Informally, a Euclidean space is supposed to be an idealization of something like a giant sheet of paper with an origin marked in pencil, or interstellar space with an origin marked by a certain star. If you're in the habit of carrying around a tape measure, a space like this has a natural metric, and you can turn it into a vector space in the obvious way (using the metric to define scalar multiplication and the parallelogram rule to define addition).

From this point of view, Euclidean space clearly has no natural basis, because if you're stranded on a giant sheet of paper, or floating in interstellar space, there's no natural set of "special" directions.

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Unfortunately, I don't know offhand how to formalize this argument. My guess is that you would start with Hilbert's axioms for Euclidean $n$-space, and choose an arbitrary point to be the origin. Hartshorne mentions in Geometry: Euclid and Beyond that in Hilbert's framework, the congruence classes of line segments naturally become the positive elements of an ordered field, which is of course isomorphic to $\mathbb{R}$. Choosing an arbitrary congruence class of line segments to be the "unit segments," you get a metric on your space. You can then turn the set of points into a vector space, using the metric to define scalar multiplication and the parallelogram rule to define addition (just like before, but now rigorously). It seems obvious to me that this vector space will have no natural basis.

Answer 14

Let $K$ be a field, let $S$ be a set, and consider the $K$-vector space $\operatorname{Map}(S,K)$ of all functions from $S$ to $K$.

When $S$ is finite, $\operatorname{Map}(S,K)$ has a natural basis: for each $x \in S$, let $\delta_x$ be the function which takes $1$ at $x$ and $0$ otherwise. However, when $S$ is infinite, these "Dirac" functions span only the set of finitely nonzero functions. In this case, the idea that there is no "natural basis" can probably be stated and proven in categorical language. (If you wish to do so as an addendum to this answer, please feel free!)

Note that one may also look at this construction in terms of the distinction between direct products and direct sums.

Answer 15

This example generalizes some of the others already mentioned: Take an infinite family of vector spaces $(V_i)_{i \in I}$. Now what about $\prod_{i \in I} V_i$, can you write down a basis?

Also, it is easy to construct an infinite multilinear tensor product $\bigotimes_{i \in I} V_i$. However, writing down a basis is equivalent to find a set of representatives of $\prod_{i \in I} V_i \setminus \{0\} / \sim$, where $\sim$ identifies families of elements, which differ only at finitely many indices. And this cannot be done explicitely.

Answer 16

Cohomology with coefficients in $\mathbb{Q}$.

Answer 17

I'll use this question as an occasion to advertise to mathematicians a interesting piece of terminology commonly used in condensed matter physics: ``degenerate ground state manifold''. In mathematical terms, this translates to ``eigenspace for the lowest eigenvalue of the Hamiltonian, whose dimension is $\ge2$, and that does not have a preferred basis of eigenvectors''.

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Let me analyze the individual terms of the phrase:

State: Here, a ``state'' is an eigenvector of the Hamiltonian.

ground: A state is a ``ground state'' if its corresponding eigenvalue (=energy) is the lowest.

degenerate: Generically, the eigenspaces corresponding to the various eigenvalues of the Hamiltonian will be one-dimensional . When that doesn't happen, an energy level is called degenerate. The word ``degeneracy'' is then used to refer to its dimension.

Manifold: Here, physicists use the term ``manifold'' here because the lowest energy eigenspace does not have a natural basis (or has more than one natural basis that one could write down). This is somewhat similar to the use of the term ``manifold'' by mathematicians: a manifold is a space on which one does not have preferred choices of coordinates (or one has multiple choices of coordinate systems).