Functional Analysis Question? I have a question about functional analysis. I know in finite dimensional space $\mathbb{C}^n$, all bases have the same cardinality. However let us consider $L^{2}[-\pi,\pi]$ which has TWO bases $\delta_{y}(x)$ for $y \in [-\pi,\pi]$ and fourier series $\{ e^{i n x} \}_{n=-\infty}^{\infty}$. One is countable, the other is not Voila! Same spaces different (cardinality) bases? Please give reference to how this can happen? Has there been any research on this (recently)? 
 A: If $\delta_y$ denotes the Dirac delta "function" then no, those don't give a basis for $L^2$; they're not even elements of $L^2$.
In any case, any two bases have the same cardinality. Except that there are different flavors of "basis" here.
"Basis" could mean basis in the sense of straight linear algebra; any element is a unique (finite) linear combination of the basis vectors. These are sometimes called "Hamel bases". Any two Hamel bases for a vector space have the same cardinality.
In a Hilbert space there's also the notion of "orthonormal basis". That's not a Hamel basis. For example those exponentials you mention give an orthonormal basis for $L^2([-\pi,\pi])$. It's a fact that any two orthonormal bases for a Hilbert space have the same cardinality.
There's also the notion of "Schauder basis" for a Banach space. A Schauder basis is countable, by definition.
At least by the typical definition. A person could say that $B$ is an "extended Schauder basis" for the Banach space $X$ if for any $x\in X$ there exist unique scalars $x_b$ for every $b\in B$ such that all but countably many $x_b$ are $0$ and $x=\sum_{b\in B} x_b b$. I don't know for a fact that any two such things must have the same cardinality, but I'd be astonished if it were not so. 
(Here when I say $x=\sum_{b\in B} x_bb$ I mean that for every $\epsilon>0$ there exists a finite set $E\subset B$ such that if $F$ is any finite subset of $B$ with $E\subset F$ then $$\left|\left|x-\sum_{b\in F}x_bb\right|\right|<\epsilon.$$Which is to say $x$ is the limit of a certain net in $X$.)
A: If you consider continuous functions on $[-\pi,\pi]$ with the $L^{2}$ norm, then it is possible to think about point values of the functions. This is an inner product space, but the space is not so useful because it is not complete, meaning that there are Cauchy sequences with no limit. This is analogous to working only with the rational numbers without allowing real numbers. Important tools such as the Spectral Theorem, orthogonal projection, orthogonal decomposition, etc. are not there in incomplete spaces.
The completion of the rational numbers, i.e., the smallest extended space where all Cauchy sequences converge, is the real numbers. In this space, all Cauchy sequences converge.
The completion of continuous functions under the $L^{2}$ norm gives you the Hilbert space of Lebesgue measurable functions $f$ for which $\int |f|^{2}dx < \infty$. However, because $\int |f-g|^{2}dx=0$ if $f=g$ except on a set of Lebesgue measure $0$, the integral cannot distinguish between two such functions if they differ on a set of measure $0$. So an element $f$ of this space must be treated as the set of all functions $g$ that are equal a.e. to $f$. This is an equivalence class $[f]$. The problem with this is that there is no definite value for $[f]$ at $x$ because there are functions in that class that take on every possible value in the scalar field, and there is typically no way to select out one element in that class over another. The integral of $[f]$ is well-defined over any Lebesgue measurable set $E$ because $\int_{E}fdx=\int_{E}gdx$ for all $g \in [f]$. But a $\delta$ function makes no sense at all because the elements of $[f]$ do not have just one value at any given point.
Mathematical foundations for Quantum were laid out by John von Neumman, and his work is still interesting to read and consistent. But it's beyond the typical undergraduate level. It turns out that nearly all of the rigorous Mathematics needed for Quantum was there before Quantum arrived on the scene, but people latched onto Dirac's point of view because of it's intuitive nature, and the great achievements of Dirac. As von Neumann wrote, 

"The method of Dirac, mentioned above, (and this is overlooked today in a great part of the quantum mechanical literature, because of the clarity of the theory) in no way satisfies the requirements of mathematical rigor--not even if these are reduced in a natural and proper fashion to the extent common elsewhere in theoretical physics. ... It should be emphasized that the correct structure need not consist in a mathematical refinement and explanation of the Dirac method, but rather that it requires a procedure differing from the very beginning, namely, the reliance on the Hilbert theory of operators."

Dirac's treatment of delta functions, because of "the clarity and elegance of the theory," has seduced many people into considering logically inconsistent ideas. One of those ideas is that delta functions form a basis of Hilbert space $L^{2}[a,b]$ or $L^{2}(\mathbb{R})$. That's just not true. Dirac's treatment was modeled after the classical eigenfunction expansions associated with Sturm-Liouville ODEs, and that model is well-correlated with what Dirac writes, but even there his use of $\delta$ functions is incorrect, though compelling. But Dirac's formalism helps you understand the correct formalism by giving it some intuition. Just don't take the quantum use of $\delta$ functions too seriously. You can take Dirac's work literally or you can take it seriously, but not both.
