A Hilbert space is a complete inner product space, meaning it is a vector space equipped with an inner product that is also complete with respect to the norm induced by this inner product. Here’s the formal definition: ## Definition of a Hilbert Space Let $H$ be a vector space over the field $\mathbb{R}$ (real numbers) or $\mathbb{C}$ (complex numbers). $H$ is called a Hilbert space if it satisfies the following conditions: 1. **Inner Product**: $H$ has an inner product $\langle \cdot, \cdot \rangle : H \times H \rightarrow \mathbb{R}$ (or $\mathbb{C}$ for complex Hilbert spaces) such that for all $x, y, z \in H$ and $\alpha \in \mathbb{R}$ (or $\mathbb{C}$), the following properties hold: - **Conjugate Symmetry**: $\langle x, y \rangle = \overline{\langle y, x \rangle}$, where $\overline{\langle y, x \rangle}$ denotes the complex conjugate. - **Linearity in the First Argument**: $\langle \alpha x + y, z \rangle = \alpha \langle x, z \rangle + \langle y, z \rangle$. - **Positive-Definiteness**: $\langle x, x \rangle \geq 0$ and $\langle x, x \rangle = 0$ if and only if $x = 0$. 2. **Norm**: The inner product defines a norm on $H$ by $\|x\| = \sqrt{\langle x, x \rangle}$ for all $x \in H$. 3. **Completeness**: $H$ is complete with respect to the norm $\|\cdot\|$. That is, every [[Cauchy Sequence]] $\{x_n\}$ in $H$ (a sequence where $\|x_n - x_m\| \to 0$ as $n, m \to \infty$) has a limit in $H$. ## Explanation of [[Completeness]] Completeness means that if a sequence of vectors $\{x_n\}$ gets arbitrarily close together, it converges to a vector within $H$. This property is essential for Hilbert spaces, distinguishing them from pre-Hilbert spaces (which have inner products but are *not necessarily complete*). ## Examples of Hilbert Spaces - **Euclidean Space $\mathbb{R}^n$ or $\mathbb{C}^n$**: These are finite-dimensional Hilbert spaces with the standard dot product as the inner product. - **Space of Square-Integrable Functions $L^2$**: The set of complex-valued functions $f$ on a domain $D$ such that $\int_D |f(x)|^2 \, dx < \infty$ forms an infinite-dimensional Hilbert space with the inner product $\langle f, g \rangle = \int_D f(x) \overline{g(x)} \, dx$. - **Sequence Space $\ell^2$**: The set of infinite sequences $\{a_n\}$ such that $\sum_{n=1}^{\infty} |a_n|^2 < \infty$ is a Hilbert space with the inner product $\langle \{a_n\}, \{b_n\} \rangle = \sum_{n=1}^{\infty} a_n \overline{b_n}$. Hilbert spaces are foundational in functional analysis, physics (especially quantum mechanics), and many areas of mathematics and engineering.