We know that a vector is an element of a vector space that does not depend by the basis that is chosen to represent it. But, in a finite dimensional vector space, when we want define a specific vector, we usually gives its components with respect to some basis. Than we can change the basis and the components of the vector change in a suitable manner such that the vector remain, geometrically, the same.
But I dont know a simple way to define a vector without give his components with respect to some basis.
The situation seems different in an infinite dimensional vector space, as a space of function. In this case ve can define a vector as a function that satisfies some conditions ( e.g. n $L^2$ function), and this does not require a basis, so, doing this, we have e definition of the vector (function) that does depend from a basis. If we want, we can now use a basis and define the components of the given function as coefficients of a series development ( a generalized Fourier expansion) and these coefficients are the analogous of the components of a vector in a finite dimensional space.
So it seems that in an infinite dimensional vector space we can define the vectors in an ''intrinsic'' way, without using a basis, but this is not possible (really ?) in a finite dimensional space.
It's this really true? and there is some motivation?
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$\begingroup$Let $V$ be a vector space. Let $x\in V$. Then $x$ is a vector.
More seriously, you're asking how we can specify vectors in a vector space without 'choosing a basis' in some way. But let me ask you this: how can we specify a vector space at all without making a similar sort of choice?
Take finite dimensional vector spaces over $\mathbb R$, for instance. How do we know that a real vector space of dimension $2$ exists? Well, that's easy - take $\mathbb R^2$. Note: $\mathbb R^2$ is defined to be the set of all pairs $(x,y)$, where $x$ and $y$ are real numbers, together with the usual (pointwise) addition and scalar multiplication rules. It is a concrete object that we construct in order to satisfy the axioms of the abstract notion of 'real vector space of dimension $2$'.
Now it's true that, in the category of real vector spaces, there's no difference between $\mathbb R^2$ or $\mathbb C$ or the space of Fibonacci-type sequences or the space of real-valued functions of the form $ae^x+be^{-x}$. But each one of these examples comes with some recipe for specifying points built in: $\mathbb R^2$, for example, has the canonical basis $(1,0), (0,1)$.
Now it would be very interesting if we could somehow define a truly abstract $2$-dimensional real vector space that came with no automatic basis attached to it. But no one has been able to do so - just try try and do the following:
Prove that there exists a real vector space of dimension $2$ without exhibiting an example that has a natural basis or a natural way of specifying vectors.
The reason you can't do that is that the vector space has to have an underlying set. In the language of mathematics, it is impossible to specify a set without being able to tell what its elements are.
Now, to answer your question, it is possible to have vector spaces where there is no canonical choice of basis, and we will still have a way of specifying vectors. Take one of my examples from above: the space of all real-valued functions of the form $ae^x+be^{-x}$, where $a,b$ are real numbers. Now one choice of basis is $e^x,e^{-x}$. But equally good are $\sinh x,\cosh x$, or $2e^x-e^{-x}, e^{-x}$, or any one of a number of examples. The function $\sinh x$ is a member of that vector space, but I don't need to specify its coordinates with respect to some basis.
Or take the complex numbers. They form a $2$-dimensional real vector space. But we will happily talk about $\sqrt{i}$ or $e^{2\pi i/5}$ without specifying their real or imaginary parts. We have a standard basis ($1,i$), but there's no reason to use it if we don't have to.
In the end, though, $\mathbb R^2$ is a much more convenient model for the theory of $2$-dimensional real vector spaces than any of the other examples, so we tend to use it. From a philosophical perspective, I'd prefer to regard writing an element of $\mathbb R^2$ as $(a,b)$ as just being the way we refer to elements of the set $\mathbb R^2$, rather than as giving the coordinates of the point with respect to some basis, but perhaps that's just personal preference.
Now, it is true that there is some notion of 'abstract $2$-dimensional real vector space where there is no choice of basis' in mathematics, but it's not an example of a vector space like the ones above. Instead, it's the collection of all examples of $2$-dimensional real vector spaces (if you want to be fancy, together with the linear maps between them). And this brings us back to the slightly facetious point I made at the top: it's perfectly fine to say 'let $V$ be a $2$-dimensional real vector space'. We know that, if we had to, we could give an example of such a space, just to prove that what we're doing isn't completely vacuous. But we don't have to, and if we did it would spoil the purity of what we're doing.
Then, if we want to specify a 'vector' inside $V$, we just say, 'let $x\in V$'. And we're done.
$\endgroup$ $\begingroup$An example: The solutions of the linear second order ODE $$y''-5y'+6y=0$$ form a two-dimensional vector space ${\cal L}$ of $C^\infty$-functions $f:\>{\mathbb R}\to{\mathbb R}$. The initial conditions $$y(0)=3,\quad y'(0)=-7$$ single out a particular element $f\in{\cal L}$ without reference to any chosen basis of ${\cal L}$.
$\endgroup$ $\begingroup$There are finite dimensional vector spaces whose elements we would not usually express as tuples of numbers. For instance, $\mathcal{P}_2$, the space of polynomials of degree at most $2$ is a real vector space of dimension $3$. However, we would not usually pick a basis for this vector space and thus give an explicit isomorphism to $\mathbb R^3$. Instead, we would write the vectors as polynomial functions, e.g. $p(x)=x^2+2$.
Another example would be the space of $2\times 3$ matrices, a $6$ dimensional vector space, whose elements we would usually write as matrices, not as tuples of numbers.
More broadly speaking, a vector space $V$ is any nonempty set in which we have defined addition and multiplication by scalars. At this level of generality, we have no idea what the elements of $V$ look like, or where they come from. We simply write $v\in V$ for some vector in this vector space. When we do this, we do not have a basis for $V$ in mind.
By the way, you should keep in mind the difference between a Hilbert basis of a Hilbert space (like $L^2$) and a vector space basis for that space. By definition, a vector space basis has the property that every element of the vector space can be written as a finite linear combination of elements of the basis (no infinite series allowed!). Thus, a vector space basis for $L^2$ will necessarily be much larger than a Hilbert basis for that space. In fact, to know such a v.s. basis exists, we need to use the axiom of choice!
$\endgroup$ $\begingroup$Finite Example
Question: How do you define $\mathbb{R}^N$?
Sloppy answer: $x=\sum_{n=1}^Nx_ne_n$
Correct answer: $x=(x_1,\ldots,x_N)$
Formal answer: $x:\{1,\ldots,N\}\to\mathbb{R}$
(That is your basis-free representation.)
Infinite Example
Question: How do you define $\ell^2(S)$?
Answer: $x:S\to\mathbb{C}:\quad\ldots$
(That is the analogue to the above.)
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