Hilbert Spaces 1 | Introductions and Cauchy-Schwarz Inequality [dark version]

Hilbert spaces are built on one central ingredient: an inner product that turns a vector space into a geometric setting where lengths, angles, and orthogonality make sense. The series frames Hilbert spaces as the backbone of functional analysis and highlights why they matter for both theory and applications—especially once infinite-dimensional problems enter the picture.

The groundwork starts with an inner product on a vector space X over a fixed field F (either real numbers R or complex numbers C). The inner product is a function ⟨·,·⟩ that takes two vectors from X and returns a scalar in F. It must be positive definite: ⟨x,x⟩ is always nonnegative, and it equals 0 only when x is the zero vector. It also must be linear in the second argument, meaning ⟨x, y1 + y2⟩ = ⟨x,y1⟩ + ⟨x,y2⟩ and ⟨x, αy⟩ = α⟨x,y⟩. Finally, it must satisfy conjugate symmetry: swapping the arguments leaves the value unchanged in the real case, but introduces complex conjugation in the complex case. That last point forces a key asymmetry: in complex spaces, the first argument behaves conjugate-linearly rather than linearly under scalar multiplication.

With those axioms in place, the series moves to the Cauchy–Schwarz inequality, a cornerstone result that links the inner product to a bound on “correlation” between vectors. For any vectors x and y, it gives

|⟨x,y⟩|² ≤ ⟨x,x⟩⟨y,y⟩.

The proof uses a standard trick: when y ≠ 0, consider the vector x − s y where s is chosen as ⟨x,y⟩ / ⟨y,y⟩. Positive definiteness ensures ⟨x − s y, x − s y⟩ ≥ 0. Expanding this expression requires careful bookkeeping of linearity in the second slot and conjugation in the first slot (the order of arguments matters in the complex case). After cancellations, the inequality emerges by rearranging terms and multiplying by ⟨y,y⟩.

Cauchy–Schwarz then immediately yields a norm from the inner product: define ||x|| = sqrt(⟨x,x⟩). The series notes that this norm satisfies the norm axioms, including the triangle inequality, which can be derived using Cauchy–Schwarz. At that point, the definition of a Hilbert space becomes clear: an inner product space is a Hilbert space when it is complete with respect to this induced norm (a Banach space condition). Completeness is what prevents “Cauchy sequences” from converging to something outside the space, making Hilbert spaces the natural setting for infinite-dimensional analysis.

The takeaway is a clean pipeline: inner product axioms → Cauchy–Schwarz → a norm → completeness → Hilbert space. That chain explains why Hilbert spaces are both mathematically structured and practically useful for studying operators, projections, and spectral behavior later in the course.