Hilbert Spaces 2 | Examples of Hilbert Spaces [dark version]

Hilbert spaces are built from vector spaces plus an inner product, and the key practical takeaway is that many familiar “function” and “sequence” settings become Hilbert spaces once the right notion of completeness and inner-product geometry is in place. The construction matters because it turns abstract linear algebra ideas—like norms, orthogonality, and convergence—into a consistent framework for both finite and infinite-dimensional problems.

A Hilbert space is defined as a complete inner product space: start with a vector space X over a field F (either ℝ or ℂ), equip it with an inner product ⟨·,·⟩ that is positive definite, linear in the second argument, and conjugate symmetric, and then define the induced norm by \|x\| = sqrt(⟨x,x⟩). Completeness refers to the fact that every Cauchy sequence in the induced norm converges within the space. In finite dimensions, completeness is automatic, which is why the series shifts attention to infinite-dimensional examples where completeness is nontrivial.

The first major examples come from standard linear algebra. On ℂ^N (and similarly on ℝ^N), choosing the usual inner product gives a Hilbert space; the same remains true if one uses other valid inner products. The infinite-dimensional story begins with sequence spaces: \ell^2, the space of square-summable sequences (x_n) with \sum_{n=1}^\infty |x_n|^2 < ∞. For complex sequences, the inner product is defined by \langle y,x\rangle = \sum_{n=1}^\infty \overline{y_n} x_n, and convergence of this series is guaranteed via Hölder’s inequality. This is the infinite-dimensional analogue of the familiar dot product on finite vectors.

The construction then generalizes from sequences to functions using measure theory. Given a measure space (Ω, A, μ), one forms L^2(Ω) as the set of measurable functions f: Ω → ℂ that satisfy \int_Ω |f|^2 dμ < ∞. The induced “norm candidate” \sqrt{\int_Ω |f|^2 dμ} can fail to be a true norm because distinct functions can have integral zero (for instance, functions that differ only on sets of measure zero). The fix is to quotient out those problematic functions: define N as the set of functions with zero norm, and form the quotient space L^2 / N. Elements are equivalence classes, where two functions are equivalent if their difference lies in N. On these equivalence classes, the norm becomes well-defined and positive definite.

With the quotient in place, an inner product is defined on equivalence classes by integrating the product of representatives: \langle g,f\rangle = \int_Ω \overline{g} f\, dμ. This yields a genuine Hilbert space, including the classic case L^2(ℝ) with respect to Lebesgue measure. The payoff is conceptual: Hilbert spaces provide a geometry from inner products, making notions like orthogonality meaningful even in highly abstract settings—setting up the next step in the series.