Hilbert Spaces

Welcome to the study of Hilbert spaces, a fundamental concept in functional analysis with wide-ranging applications in quantum mechanics, signal processing, and partial differential equations. A Hilbert space is a complete inner product space. Let's break down what that means and explore its key components.

Inner Product Spaces

An inner product space is a vector space endowed with an operation called an inner product. This inner product takes two vectors and returns a scalar, satisfying certain properties. For a vector space $V$ over the field $\mathbb{C}$ (complex numbers) or $\mathbb{R}$ (real numbers), the inner product, denoted by $\langle \cdot, \cdot \rangle$, must satisfy:

• Conjugate Symmetry: $\langle x, y \rangle = \overline{\langle y, x \rangle}$ for all $x, y \in V$. If the field is $\mathbb{R}$, this simplifies to symmetry: $\langle x, y \rangle = \langle y, x \rangle$.
• Linearity in the First Argument: $\langle ax + by, z \rangle = a\langle x, z \rangle + b\langle y, z \rangle$ for all $x, y, z \in V$ and scalars $a, b$.
• Positive-definiteness: $\langle x, x \rangle \ge 0$ for all $x \in V$, and $\langle x, x \rangle = 0$ if and only if $x$ is the zero vector.

From these properties, we can deduce linearity in the second argument for complex spaces: $\langle x, ay + bz \rangle = \overline{a}\langle x, y \rangle + \overline{b}\langle x, z \rangle$.

The inner product allows us to define the norm of a vector, which is a measure of its length: $\|x\| = \sqrt{\langle x, x \rangle}$. This norm induces a distance (or metric) $d(x, y) = \|x - y\|$, making the inner product space a metric space.

Completeness

A complete metric space is one where every Cauchy sequence converges to a limit within the space. A Cauchy sequence is a sequence where the terms get arbitrarily close to each other as the sequence progresses. Completeness is a crucial property that ensures the existence of limits, which is essential for many analytical operations, like solving differential equations or performing Fourier analysis.

Hilbert Space Definition

A Hilbert space $H$ is a vector space equipped with an inner product that is complete with respect to the norm induced by the inner product.

Examples of Hilbert Spaces:

• Finite-dimensional Euclidean Spaces: $\mathbb{R}^n$ and $\mathbb{C}^n$ with the standard dot product $\langle x, y \rangle = \sum_{i=1}^n x_i \overline{y_i}$. These are finite-dimensional and thus complete.
• Sequence Spaces: The space $l^2$ of square-summable complex sequences, i.e., sequences $\{x_n\}_{n=1}^\infty$ such that $\sum_{n=1}^\infty |x_n|^2 < \infty$. The inner product is $\langle x, y \rangle = \sum_{n=1}^\infty x_n \overline{y_n}$. This space is complete.
• Function Spaces: The space $L^2(\Omega)$ of square-integrable functions on a domain $\Omega$, i.e., functions $f$ such that $\int_\Omega |f(x)|^2 dx < \infty$. The inner product is $\langle f, g \rangle = \int_\Omega f(x) \overline{g(x)} dx$. This space is also complete.

Orthonormal Bases in Hilbert Spaces

In Euclidean geometry, we often use orthonormal bases (like the standard basis vectors $(1,0)$ and $(0,1)$ in $\mathbb{R}^2$) to represent vectors. Hilbert spaces generalize this concept.

Orthogonality and Orthonormality

Two vectors $x, y$ in an inner product space are orthogonal if $\langle x, y \rangle = 0$. A set of vectors $\{e_i\}$ is orthogonal if $\langle e_i, e_j \rangle = 0$ for all $i \neq j$.

A set of vectors $\{e_i\}$ is orthonormal if it is orthogonal and each vector has a norm of 1, i.e., $\langle e_i, e_i \rangle = \|e_i\|^2 = 1$ for all $i$.

Existence of Orthonormal Bases

A key theorem in Hilbert space theory states that every separable Hilbert space (a Hilbert space with a countable dense subset) has a countable orthonormal basis.

Gram-Schmidt Orthonormalization Process: This is a constructive method to obtain an orthonormal set from any linearly independent set of vectors in an inner product space.

Let $\{v_1, v_2, v_3, \dots\}$ be a linearly independent set of vectors. We construct an orthonormal set $\{e_1, e_2, e_3, \dots\}$ as follows:

1. $u_1 = v_1$
2. $e_1 = u_1 / \|u_1\|$ (if $v_1 \neq 0$)
3. $u_2 = v_2 - \langle v_2, e_1 \rangle e_1$
4. $e_2 = u_2 / \|u_2\|$ (if $u_2 \neq 0$)
5. $u_3 = v_3 - \langle v_3, e_1 \rangle e_1 - \langle v_3, e_2 \rangle e_2$
6. $e_3 = u_3 / \|u_3\|$ (if $u_3 \neq 0$)
7. In general, for $n > 1$: $u_n = v_n - \sum_{k=1}^{n-1} \langle v_n, e_k \rangle e_k$
8. And $e_n = u_n / \|u_n\|$ (if $u_n \neq 0$)

Properties of Orthonormal Bases

If $\{e_i\}_{i \in I}$ is an orthonormal basis for a Hilbert space $H$, then any vector $x \in H$ can be uniquely represented as a (possibly infinite) linear combination:

$x = \sum_{i \in I} c_i e_i$

where the coefficients $c_i$ are given by the inner product:

$c_i = \langle x, e_i \rangle$

This is known as the Fourier series expansion of $x$ with respect to the orthonormal basis $\{e_i\}$. The coefficients $c_i$ are called the Fourier coefficients.

The Parseval's Identity holds for any $x, y \in H$:

$\langle x, y \rangle = \sum_{i \in I} c_i \overline{d_i}$

where $c_i = \langle x, e_i \rangle$ and $d_i = \langle y, e_i \rangle$.

In particular, for the norm of $x$:

$\|x\|^2 = \langle x, x \rangle = \sum_{i \in I} |c_i|^2 = \sum_{i \in I} |\langle x, e_i \rangle|^2$

Conjugate Space (Dual Space)

The conjugate space, or dual space, of a normed vector space $X$, denoted by $X^*$, is the set of all continuous linear functionals on $X$. A linear functional $f$ is a linear map from $X$ to its underlying scalar field ( $\mathbb{R}$ or $\mathbb{C}$).

For a Hilbert space $H$, the situation is particularly elegant due to the Riesz Representation Theorem.

Riesz Representation Theorem

The Riesz Representation Theorem states that for every continuous linear functional $f$ on a Hilbert space $H$, there exists a unique vector $y \in H$ such that $f(x) = \langle x, y \rangle$ for all $x \in H$.

This theorem establishes a one-to-one correspondence between the vectors in $H$ and the continuous linear functionals on $H$. This means that the dual space $H^*$ of a Hilbert space $H$ is isometrically isomorphic to $H$ itself.

$H^* \cong H$

The correspondence is given by $y \mapsto f_y$, where $f_y(x) = \langle x, y \rangle$.

The norm of the functional $f_y$ is $\|f_y\| = \sup \{|f_y(x)| : \|x\| = 1\} = \sup \{|\langle x, y \rangle| : \|x\| = 1\}$. By the Cauchy-Schwarz inequality, $|\langle x, y \rangle| \le \|x\|\|y\|$, so $|\langle x, y \rangle| \le \|y\|$ when $\|x\| = 1$. This supremum is attained when $x = y/\|y\|$, giving $\|f_y\| = \|y\|$.

Adjoint of an Operator

In linear algebra, the transpose of a matrix $A$, denoted $A^T$, has the property $\langle Ax, y \rangle = \langle x, A^T y \rangle$. In Hilbert spaces, we generalize this concept to the adjoint operator.

Definition of the Adjoint Operator

Let $H$ be a Hilbert space and $T: H \to H$ be a bounded linear operator. The adjoint operator of $T$, denoted by $T^*$, is a bounded linear operator $T^*: H \to H$ such that for all $x, y \in H$:

$\langle Tx, y \rangle = \langle x, T^* y \rangle$

Existence and Uniqueness

The existence and uniqueness of the adjoint operator for a bounded linear operator on a Hilbert space is guaranteed by the Riesz Representation Theorem. For a fixed operator $T$, consider the functional $g_y(x) = \langle Tx, y \rangle$. This is a linear functional of $x$. By Riesz's theorem, there exists a unique vector $z \in H$ such that $g_y(x) = \langle x, z \rangle$. We define $T^*y = z$.

We need to show $T^*$ is linear and bounded. Linearity follows from the properties of the inner product and the uniqueness part of Riesz's theorem. Boundedness also follows from the Riesz Representation Theorem, as $\|T^*y\| = \|g_y\| = \sup_{x \neq 0} \frac{|\langle Tx, y \rangle|}{\|x\|}$.

Properties of the Adjoint Operator

Let $T, S$ be bounded linear operators on a Hilbert space $H$, and let $c$ be a scalar. Then:

• $(T^*)^* = T$
• $(T + S)^* = T^* + S^*$
• $(cT)^* = \overline{c}T^*$
• $(TS)^* = S^*T^*$
• If $T$ is invertible, then $(T^{-1})^* = (T^*)^{-1}$

Self-Adjoint Operators

A bounded linear operator $T$ is called self-adjoint (or Hermitian) if $T^* = T$.

For a self-adjoint operator $T$:

• $\langle Tx, y \rangle = \langle x, Ty \rangle$ for all $x, y \in H$.
• The eigenvalues of a self-adjoint operator are real.
• If $T$ is self-adjoint, then $\langle Tx, x \rangle$ is real for all $x \in H$.

Self-adjoint operators are crucial in quantum mechanics, where observables (like position, momentum, energy) are represented by self-adjoint operators.

Normal Operators

A bounded linear operator $T$ is called normal if $TT^* = T^*T$.

Self-adjoint operators are a special case of normal operators ($T^*=T \implies TT^* = T^2 = T^*T$). Unitary operators are also normal. Normal operators have a rich spectral theory and are diagonalizable by a unitary operator.

Projections

In geometry, a projection is a map that maps a point onto another point, typically onto a line or a plane. In Hilbert spaces, projections are operators that generalize this idea.

Definition of a Projection Operator

A bounded linear operator $P$ on a Hilbert space $H$ is called a projection if it is idempotent, meaning $P^2 = P$.

If $P$ is a projection, its range, $M = P(H) = \{Px : x \in H\}$, is a subspace of $H$. Its null space, $N = \ker(P) = \{x \in H : Px = 0\}$, is also a subspace.

For any $y \in M$ and $z \in N$, we have $y = Px$ for some $x$, and $Pz = 0$. Then $\langle y, z \rangle = \langle Px, z \rangle$.

Orthogonal Projections

A projection $P$ is called an orthogonal projection if its range $M$ and null space $N$ are orthogonal complements. This means $H = M \oplus N$ (direct sum) and $\langle m, n \rangle = 0$ for all $m \in M$ and $n \in N$.

An equivalent condition for an orthogonal projection is that it must be self-adjoint. So, an operator $P$ is an orthogonal projection if and only if $P^2 = P$ and $P^* = P$.

If $P$ is an orthogonal projection onto a subspace $M$, then for any $x \in H$, $Px$ is the unique vector in $M$ that is closest to $x$.

Properties of Orthogonal Projections

Let $P$ be an orthogonal projection onto a subspace $M$.

• $P^2 = P$
• $P^* = P$
• The range of $P$ is $M$, and the null space of $P$ is $M^\perp$ (the orthogonal complement of $M$).
• For any $x \in H$, $\|Px\| \le \|x\|$. This means $\|P\| \le 1$ (unless $M = \{0\}$). If $P$ is not the zero projection, then $\|P\| = 1$.
• $I - P$ is also an orthogonal projection, onto the subspace $M^\perp$.

Projections onto Orthonormal Sets

Let $\{e_i\}_{i \in I}$ be an orthonormal set in a Hilbert space $H$. For any $x \in H$, we can define the projection onto the subspace spanned by a finite subset of these vectors.

If $I$ is finite, say $I = \{1, 2, \dots, n\}$, then the orthogonal projection $P_M$ onto the subspace $M = \text{span}\{e_1, \dots, e_n\}$ is given by:

$P_M x = \sum_{i=1}^n \langle x, e_i \rangle e_i$

This operator satisfies $P_M^2 = P_M$ and $P_M^* = P_M$.

If $H$ is separable and $\{e_i\}_{i=1}^\infty$ is a countable orthonormal basis, then the projection onto the entire space $H$ is the identity operator:

$Ix = x = \sum_{i=1}^\infty \langle x, e_i \rangle e_i$

The operator $P_M$ is the limit of finite projections as more basis vectors are included.

Summary of Key Concepts

Hilbert Space: A complete inner product space.

Orthonormal Basis: A set of mutually orthogonal unit vectors that spans the space. Allows unique representation of any vector as a Fourier series.

Conjugate Space ($H^*$): The space of continuous linear functionals. For Hilbert spaces, $H^* \cong H$ via the Riesz Representation Theorem.

Adjoint Operator ($T^*$): Satisfies $\langle Tx, y \rangle = \langle x, T^* y \rangle$. Essential for studying properties like self-adjointness.

Self-Adjoint Operator: $T^* = T$. Eigenvalues are real.

Projection Operator: $P^2 = P$. Maps vectors onto a subspace.

Orthogonal Projection: $P^2 = P$ and $P^* = P$. Maps vectors onto a subspace orthogonally.