Sometimes the way authors write a book makes you wonder if they are at their wits' end: 'A vector space with an inner product is an inner product space!' I am not sure if I have gone crazy but to me it is obvious that if you have a vector space wrapped up in a nice gift basket and sent off to your address from Amazon, then the inner product comes with the vector space for free. After all, the inner product is obtained from any two vectors in a vector space, so I just can't understand why a vector space cannot be pre-equipped with an inner product, that is, why the hell isn't a vector space and an inner product space the same thing?
Because the definition of vector space doesn't include an inner product. An inner product is a function mapping a pair of vectors to an element of the underlying field; until you have defined such a function, you do not have an inner product. The definition of vector space does not include such a function, therefore, a vector space is not necessarily an inner product space.
Given any (finite dimensional) vector space there are an infinite number of possible "inner products". For example, choose any basis, [itex]\{e_1, e_2, ..., e_n\}[/itex]. We can now write two vectors, [itex]u= a_1e_1+ a_2e_2+ ...+ a_ne_m[/itex] and [itex]v= b_1e_1+ b_2e_2+ ...+ b_ne_n[/itex], written in terms of that basis. We define the inner product [itex]<u, v>= a_1b_1+ a_2b_2+ ...+ a_nb_n[/itex]. Choosing a different basis will give a different inner product. (And the "theoretical meat" of the Gram-Schmidt orthogonalization process is that, given any abstractly defined inner product there exist a basis in which that inner product is as given above. And note the "finite dimensional". There exist infinite dimensional vector spaces which do not have any inner product.
That's not true - your method will produce an inner product on a real vector space of any dimension. (The trouble, of course, is "choosing" a basis. But once you have one, you're good to go.)
Yes, they all have an inner product. The dimension, as in cardinality of a basis, is a complete invariant if all you are looking at is the vector space structure, so there's not much variety there. However, if you have a norm, not all norms are induced by inner products and that's probably what he was thinking. For example, L^p is not an inner product space for p not equal to 2 (check the parallelogram identity).
A vector space over a field F is defined as a triple (set, addition operation, scalar multiplication operation) that satisfies a bunch of axioms. If F=ℝ or F=ℂ, then an inner product space over F is defined as a pair (vector space over F, inner product) that satisfies a bunch of axioms. For example, when we refer to ℝ^{2} as a vector space, we're actually being sloppy. ℝ^{2} is just a set. However, if we define three functions ##A:\mathbb R^2\times\mathbb R^2\to\mathbb R^2##, ##S:\mathbb R\times\mathbb R^2\to\mathbb R^2## and ##I:\mathbb R^2\times\mathbb R^2\to\mathbb R## by $$ \begin{align} A\big((x_1,x_2),(y_1,y_2)\big) &=(x_1+x_2,y_1+y_2)\\ S\big(a,(x_1,x_2)\big) &=(ax_1,ax_2)\\ I\big((x_1,x_2),(y_1,y_2)\big) &=x_1 x_2+y_1y_2 \end{align} $$ for all ##(x_1,x_2), (y_1,y_2)\in\mathbb R^2## and all ##a\in\mathbb R##, then ##(\mathbb R^2,A,S)## is a vector space over ℝ, and if we denote that space by V, then ##(V,I)## is an inner product space over ℝ. This post may be useful.
Not quite; if you're thinking about integration, you are selecting countably-many points, and countably-many partitions. Any uncountable sum with more than countably-many non-zero terms, necessarily diverges. Just partition your uncountable support-set into sets An:={x:x>1/n}; at least one of the sets will have infinitely-many terms.
This thread's drifted a bit, but the answer to your question is that when you care about the vector space properties, there's no point adding in extraneous details. If you want to intersect two sets, there's no point mentioning that the sets are also vector spaces or groups or manifolds or anything else. If you loaded up every definition with all of its derived types, it would be incredibly confusing. A set is such and so. A group is a set with such and so. A Lie group is a group with such and so. You build up complex definitions in terms of simpler ones. If all you care about is the properties of a vector space, why add on additional properties that you don't care about?
Yes but Schauder bases are irrelevant here. The claim was that we can construct an inner product on any vector space. The argument was: (1) Choose a (Hamel) basis. (2) Every vector is a finite linear combination of vectors from this basis (even if the basis itself is infinite). (3) HallsofIvy's recipe for an inner product still works, regardless of whether the vector space is finite- or infinite-dimensional.
Right, my bad, I lost focus and was making a general statement about sums in V.Spaces. The force is back with me now.
That's true, but it's not the only issue. The other issue is that there is no canonical choice of inner product. They all have an inner product. Requiring the existence of an inner product doesn't add anything because it follows from the definition of a real or complex vector space. But which inner product? The definition isn't a vector space on which there exists an inner product. It's a vector space with some chosen inner product that you have singled out.
Am I missing something obvious here? Clearly it works for those vectors that belong to the smallest vector subspace that contains all the basis vectors (because the members of that set are linear combinations of basis vectors), but if the vector space we're talking about is the closure of that subspace, then it seems to me that it should fail.
But we're working with a (Hamel) basis for the entire vector space, so (by definition!) every vector is a finite linear combination of elements in the basis!
OK, I get it now. Thanks. It wouldn't work for an orthonormal basis (=maximal orthonormal set) for a Hilbert space, but it would work for a Hamel basis (=maximal linearly independent set). Of course, Hilbert spaces already have inner products, and there's usually no need to define another one.
To OP interesting fact might be that at least in finite dimensional case, all inner products are kinda similar, since every inner product can be written as [itex]<u, v>= k_1 a_1b_1+ k_2 a_2b_2+ ...+ k_n a_nb_n[/itex] for suitable choice of basis. In real case, this can be reduced to [itex]<u, v>= a_1b_1+ a_2b_2+ ...+ a_nb_n[/itex].