A different kind of vector space
Linear algebra gives us powerful tools for analyzing Euclidean space, but its utility doesn’t stop there. By treating functions as vectors, we can carry those familiar tools—inner products, norms, bases, orthogonal projections—into the world of analysis. That extension is what Hilbert space provides.
The shift in perspective begins with how we interpret the components of a vector. A vector in R^3 is just an ordered list: the first component is the value at index 1, the second at index 2, and so on. In that sense, a vector is simply a function whose domain is the index set {1, 2, 3} and whose codomain is the reals: v(i) gives the i-th component. Generalizing to N dimensions, we get a function from {1, ..., N} to R, and nothing stops us from letting the domain grow to the entire set of natural numbers—that is, a function from N to R.
We can go further. Allow the domain to be all of R, and our vector becomes a function from the reals to the reals. Of course, we can't list the components one by one—there are uncountably many and most don't have finite representations—but we can describe the mapping with a rule, say, f(x) = sinc(x). This "vector" lives in an infinite-dimensional space, and with careful definitions, all the machinery of vector spaces still applies.
Building up from functions
Equipped with the usual pointwise addition and scalar multiplication, the set of all functions mapping an arbitrary set 
into 
forms a vector space—a well-known fact that follows by verifying the standard axioms one by one, using the associativity, commutativity, and distributivity of the underlying field. This gives us a vector space, but to do meaningful geometry and analysis, we need more structure.
The first layer of structure is a notion of size. Switching to complex-valued functions for full generality, we say f is square integrable when
∫-∞∞ |f(x)|² dx < ∞.
The set of such functions, denoted L², is closed under addition and scalar multiplication, hence a subspace of the function space above. The integral here is an analogue of the Euclidean norm—for vectors it measures length; for functions, by analogy with physics, it is often called energy.
With square integrability in hand, we can introduce an inner product on L²:
⟨f, g⟩ = ∫-∞∞ f(x) * conj(g(x)) dx.
This is the continuous generalization of the complex vector inner product, which sums products of conjugate components over indices; here the summation becomes an integral over the x-axis. The Cauchy-Schwarz inequality in its integral form confirms that the inner product is finite for all pairs f, g in L², since |⟨f, g⟩| ≤ √(⟨f, f⟩) √(⟨g, g⟩),
and both factors are finite by square integrability. That property is precisely why L², rather than a larger function space, is the right domain for this inner product.
Defining the inner product and norm
The pairing satisfies the inner product axioms, which we won't re-derive here (standard references cover it fully). The resulting norm is
||f|| = √⟨f, f⟩
where the finiteness again comes from square integrability. Together, L² together with its inner product constitutes an inner product space, and the norm it induces gives L² its metric structure.
One wrinkle remains. Describing a space as "complete" is loaded terminology in mathematics, but in this context it is unambiguous: a space is complete if every Cauchy sequence in it converges to a point in the space. In functional-analysis lingo, it means there are no holes, so to speak. The Riesz-Fischer theorem establishes that L² has this property. Once completeness holds, the inner product space becomes a Hilbert space.
That completeness distinguishes a Hilbert space from a mere inner product space in a related, though more general, family of spaces. A Banach space is a complete space equipped with a norm, which may or may not derive from an inner product. A Hilbert space is the special case where the norm is generated by an inner product. All Hilbert spaces are Banach spaces, but not conversely.
What completeness buys: Fourier series
Fourier series neatly illustrates why all of this matters. A function's expansion in terms of sines and cosines is possible precisely because L² functions are square integrable, and its inner product determines orthogonality of basis functions. Expressing a given function as a weighted sum of basis vectors is viable because the space is complete—convergence to the function, rather than to some point outside the space, is assured. In fact, the general machinery generalizes beyond trigonometric functions: via the Gram-Schmidt process on the polynomial basis, one obtains Legendre polynomials, giving us polynomial Fourier series.
This rigorous structuring came late, historically. Fourier advanced his ideas in the early 19th century, decades before the formal analysis of these spaces; contemporary mathematicians like Lagrange pushed back for lack of rigor. The theory worked too well for too many utility-laden applications to be discarded, but consolidating it on firm mathematical ground required the later languages of functional analysis and Hilbert spaces, developed by Hilbert and others in the early 20th century.
Application: operators on wavefunctions
Quantum mechanics offers a further, very concrete use case for Hilbert space concepts: the states of a quantum particle are wavefunctions in a Hilbert space. Probabilities are derived from the inner product, and physical observables act as linear maps—operators—on that space.
In that setting, the heavyweight machinery built above—infinite-dimensional vector spaces, norms, completeness, bases—is not an exotic curiosity but the daily vocabulary of a working physicist, and its application demonstrates what can happen when the tools of finite-dimensional linear algebra are unleashed on unbounded, continuous spaces.
