Hilbert spaces are the powerhouses of functional analysis, combining completeness, norms, and inner products. They're like super-charged vector spaces that let us do amazing things with infinite dimensions and abstract functions.

These spaces are everywhere in math and physics. From quantum mechanics to signal processing, Hilbert spaces give us the tools to tackle complex problems. They're the perfect playground for studying operators, orthogonality, and approximation theory.

Hilbert Spaces

Definition of Hilbert spaces

• Hilbert spaces are complete normed vector spaces equipped with an inner product
  • Completeness ensures every Cauchy sequence converges to an element within the space
  • Normed vector spaces have a notion of distance between elements defined by a norm (Euclidean spaces, $L^p$ spaces)
  • Inner product generalizes the dot product and induces a norm $|x| = \sqrt{\langle x, x \rangle}$
• Inner product satisfies properties such as conjugate symmetry, linearity, and positive-definiteness
  • Conjugate symmetry: $\langle x, y \rangle = \overline{\langle y, x \rangle}$
  • Linearity: $\langle ax + by, z \rangle = a\langle x, z \rangle + b\langle y, z \rangle$ for scalars $a, b$
  • Positive-definiteness: $\langle x, x \rangle \geq 0$ and $\langle x, x \rangle = 0$ iff $x = 0$
• Distinguish Hilbert spaces from general inner product spaces by completeness
  • General inner product spaces may not be complete (space of polynomials with $L^2$ inner product)
  • Completeness is crucial for many important results and applications in functional analysis

Completeness of Hilbert spaces

• Prove Hilbert spaces are complete with respect to the norm induced by the inner product
  • Consider a Cauchy sequence $(x_n)$ in a Hilbert space $H$
  • Show the sequence of inner products $(\langle x_n, x_m \rangle)$ is Cauchy in the underlying field ($\mathbb{R}$ or $\mathbb{C}$)
  • Use the completeness of the field to conclude $(\langle x_n, x_m \rangle)$ converges to a limit $L$
• Define the limit element $x \in H$ using the limit $L$ and properties of the inner product
  • For any $y \in H$, define $\langle x, y \rangle := \lim_{n \to \infty} \langle x_n, y \rangle$
  • Verify that $x$ is well-defined and belongs to $H$ using the boundedness of $(x_n)$
• Show that the Cauchy sequence $(x_n)$ converges to the limit element $x$
  • Use the Cauchy-Schwarz inequality and the convergence of inner products
• Conclude that every Cauchy sequence in $H$ converges, proving completeness

Examples of Hilbert spaces

• $L^2$ spaces: square-integrable functions on a measure space $(X, \mu)$
  • Functions $f: X \to \mathbb{C}$ satisfying $\int_X |f|^2 d\mu < \infty$
  • Inner product: $\langle f, g \rangle = \int_X f \overline{g} d\mu$
  • Examples: $L^2(\mathbb{R})$ (functions on the real line), $L^2([0,1])$ (functions on the unit interval)
• $\ell^2$ spaces: square-summable sequences
  • Sequences $(x_n){n=1}^{\infty}$ satisfying $\sum{n=1}^{\infty} |x_n|^2 < \infty$
  • Inner product: $\langle (x_n), (y_n) \rangle = \sum_{n=1}^{\infty} x_n \overline{y_n}$
• Finite-dimensional Euclidean spaces $\mathbb{R}^n$ and $\mathbb{C}^n$
  • Inner product is the standard dot product: $\langle x, y \rangle = \sum_{i=1}^n x_i \overline{y_i}$
• Sobolev spaces $H^k(\Omega)$: functions with square-integrable weak derivatives up to order $k$

Cauchy-Schwarz inequality in Hilbert spaces

• Statement: $|\langle x, y \rangle| \leq |x| |y|$ for all $x, y \in H$
  • Equality holds if and only if $x$ and $y$ are linearly dependent
• Geometric interpretation: relates the angle between vectors to their lengths
  • For unit vectors, $|\langle x, y \rangle|$ is the cosine of the angle between them
  • Cauchy-Schwarz inequality bounds the absolute value of the inner product by the product of norms
• Prove the Cauchy-Schwarz inequality using the properties of the inner product
  • Consider the quadratic polynomial $p(t) = \langle x + ty, x + ty \rangle$ for $t \in \mathbb{R}$
  • Use the positive-definiteness of the inner product to show $p(t) \geq 0$ for all $t$
  • Conclude that the discriminant of $p(t)$ must be non-positive, yielding the inequality
• Applications and consequences of the Cauchy-Schwarz inequality
  • Proving the triangle inequality for the norm: $|x + y| \leq |x| + |y|$
  • Deriving uncertainty principles in quantum mechanics (Heisenberg's uncertainty principle)
  • Studying the convergence of Fourier series in $L^2$ spaces