Introduction to Hilbert Spaces

The Axioms of a Hilbert Space

At its core, a Hilbert space is a complete inner product space. This definition packs three distinct mathematical concepts into one phrase. Let's unpack them. The following video provides a great conceptual overview of these criteria.

Now, let's formalize these ideas.

1. Vector Space
A Hilbert space is first and foremost a vector space over the field of complex numbers $\mathbb{C}$ (for quantum mechanics). This means it's a collection of elements (vectors) that is closed under vector addition and scalar multiplication, satisfying familiar rules like commutativity, associativity, and distributivity. Your extensive background in physics and mathematics means these axioms are second nature.

2. Inner Product Space
The second layer of structure is the inner product, denoted as $⟨\varphi ,\psi ⟩$, which takes two vectors and produces a complex number. It is a generalization of the dot product from Euclidean geometry and must satisfy specific properties.

For a detailed list of these properties, please consult the following resource.

To summarize the physics convention, for vectors $\psi ,\varphi ,\chi$ and a scalar $c\in \mathbb{C}$:

• Conjugate Symmetry: $⟨\varphi ,\psi ⟩=⟨\psi ,\varphi {⟩}^{\ast }$
• Linearity in the second argument: $⟨\varphi ,c\psi +d\chi ⟩=c⟨\varphi ,\psi ⟩+d⟨\varphi ,\chi ⟩$
• Positive-Definiteness: $⟨\psi ,\psi ⟩\ge 0$, with $⟨\psi ,\psi ⟩=0$ if and only if $\psi$ is the zero vector.

The inner product allows us to define the norm (or length) of a vector as $\Vert \psi \Vert =\sqrt{⟨\psi ,\psi ⟩}$. A crucial result that follows directly from these axioms is the Cauchy-Schwarz inequality:

This inequality is fundamental in deriving many results in quantum mechanics, including the generalized uncertainty principle, which we will encounter in a later lesson.

3. Completeness
Completeness is the most abstract but arguably the most critical property. A space is complete if every Cauchy sequence of vectors in the space converges to a limit vector that is also in the space.

A Cauchy sequence is one where the distance between its elements, $\Vert {\psi }_n-{\psi }_m\Vert$, becomes arbitrarily small as $n$ and $m$ go to infinity. Intuitively, the sequence "bunches up" as if it's converging. Completeness guarantees that the point it's converging to actually exists within the space.

The analogy from the video is perfect: the set of rational numbers $\mathbb{Q}$ is not complete because you can construct a Cauchy sequence of rational numbers (e.g., 3, 3.1, 3.14, 3.141, ...) that converges to $\pi$, which is not a rational number. The sequence converges, but its limit lies outside the original space. A Hilbert space has no such "holes".

Key Examples of Hilbert Spaces

Understanding the abstract definition is one thing; seeing it in action is another. Let's look at the most important examples for quantum mechanics. The next video provides a concise overview.

Let's solidify the key points from these examples.

• Finite-Dimensional Space: ${\mathbb{C}}^n$
  This is the space of n-tuples of complex numbers, familiar from linear algebra. The standard inner product is $⟨\mathbf{u},\mathbf{v}⟩={\sum }_{i=1}^n{u}_i^{\ast }{v}_i$. All finite-dimensional inner product spaces are automatically complete, so they are all Hilbert spaces.

• Infinite-Dimensional Sequence Space: ${\ell }^2$
  This is the space of all infinite sequences of complex numbers $\{c_n{\}}_{n=1}^{\infty }$ such that the sum of the squares of their magnitudes is finite: ${\sum }_{n=1}^{\infty }|c_n{|}^2<\infty$. The inner product is a natural extension of the finite case: $⟨\{c_n\},\{d_n\}⟩={\sum }_{n=1}^{\infty }{c}_n^{\ast }{d}_n$. Proving that ${\ell }^2$ is complete is a non-trivial result in analysis.

• Infinite-Dimensional Function Space: $L^2(a,b)$
  This is the primary state space for a particle in one dimension. It consists of all complex-valued functions $f(x)$ on the interval $(a,b)$ that are square-integrable:

  $${\int }_a^b|f(x){|}^{2}dx<\infty$$

  The inner product is defined as:

  $$⟨f,g⟩={\int }_a^b{f}^{\ast }(x)g(x)dx$$

  As the video highlighted, verifying that $L^2$ is a Hilbert space requires careful handling of two major issues.

  1. The Norm of a Function: Can a non-zero function have zero norm? Yes. Consider a function that is zero everywhere except at a single point. Its integral $\int |f(x){|}^{2}dx$ would be zero, but the function is not the zero function. This violates the positive-definiteness axiom. The solution is to work not with individual functions, but with equivalence classes of functions, where we identify any two functions if they differ only on a set of measure zero.

  2. Completeness: How do we know that any Cauchy sequence of square-integrable functions converges to another square-integrable function? This is not at all obvious. The proof relies on a cornerstone of functional analysis: the Riesz-Fischer Theorem.

The following reading provides a more formal treatment of the $L^2$ space and its completeness.

By establishing that the space of square-integrable functions is a complete inner product space, we have confirmed that $L^2$ is a Hilbert space. This rigorous mathematical foundation ensures that the tools of linear algebra can be reliably applied to the wave functions of quantum mechanics, even in infinite dimensions.

Conclusion

In this lesson, we have formally defined the mathematical structure that hosts the states of quantum systems. Let's summarize the key points:

• A Hilbert space is a complete, complex inner product space.
• The inner product equips the space with a geometry, defining notions of length (norm) and orthogonality.
• Completeness ensures the space is continuous and well-behaved, guaranteeing that convergent sequences of states have a limit state that also belongs to the space.
• The space of square-integrable functions, $L^2$, is the canonical Hilbert space for systems described by wave functions. Its formal verification requires the concept of equivalence classes and the powerful Riesz-Fischer theorem to establish completeness.

Having defined the space, our next step is to develop a practical and powerful notation for working with the vectors (states) and inner products within it. In the next lesson, we will introduce Dirac's elegant bra-ket notation, which streamlines calculations and clarifies the relationship between a Hilbert space and its dual space.