Hilbert Spaces 2 | Examples of Hilbert Spaces

Q: Why does completeness matter for turning an inner-product space into a Hilbert space?

Completeness requires that every Cauchy sequence (with respect to the norm induced by the inner product) converges to an element inside the space. Without completeness, limits of “well-behaved” sequences might fall outside the space, breaking the geometry needed for analysis. The transcript emphasizes that a Hilbert space is a special Banach space: it is complete under the norm derived from the inner product.

Q: How is the inner product on ℓ^2 defined, and what guarantees it is finite?

For sequences x and y in ℓ^2, the inner product is ⟨y,x⟩ = ∑_{n=1}^∞ y_n* x_n, where * denotes complex conjugation. The sum converges to a finite complex number because Holder’s inequality applies to square-summable sequences, ensuring the inner product is well-defined.

Q: What goes wrong if one tries to define L^2 using only square-integrable functions directly?

The integral ∫_Ω |f|^2 dμ can equal 0 even when f is not the zero function, such as when f is nonzero only on a set of measure zero. That would violate the norm axiom that ||f|| = 0 implies f is the zero vector. The transcript resolves this by quotienting out all functions with zero norm.

Q: What exactly is the quotient construction for L^2, and how does it fix the norm?

Let N be the set of measurable functions f with ∫_Ω |f|^2 dμ = 0. Two functions are treated as equivalent if their difference lies in N, meaning they agree except on measure-zero sets. Elements of L^2 are equivalence classes, and the norm is defined using the original integral on any representative; it becomes well-defined because all zero-norm functions collapse into the same class as the zero function.

Q: How is the inner product on L^2 defined once equivalence classes are used?

Represent two equivalence classes by functions G and F. The inner product is ⟨G,F⟩ = ∫_Ω G* F dμ, with complex conjugation on the first factor. Because the space is built from equivalence classes, this integral yields the same value regardless of which representatives are chosen (the quotient removes the measure-zero ambiguity).

TL;DR

A Hilbert space is a complete inner-product space, where completeness means every Cauchy sequence converges within the space.

Briefing Cornell Notes

Briefing

Hilbert spaces are best understood as complete inner-product spaces: start with a vector space over either the real numbers or the complex numbers, equip it with an inner product that is linear (in the second argument), conjugate symmetric, and positive definite, then require completeness so every Cauchy sequence converges. That completeness turns an inner-product space into a Hilbert space, and the norm comes directly from the inner product via the square root of ⟨x,x⟩. This matters because it guarantees a stable “geometry” for infinite-dimensional problems—distances, angles, and orthogonality behave reliably.

For concrete examples, the simplest starting point is finite-dimensional Euclidean space: once an inner product is fixed on ℝ^n or ℂ^n, the result is automatically a Hilbert space. Completeness is automatic in finite dimensions, so the real work begins in infinite-dimensional settings.

A classic infinite-dimensional example is ℓ^2, the space of square-summable sequences. Elements are sequences (x_n) indexed by natural numbers with values in ℂ (or ℝ in the real case) such that the series ∑_{n=1}^∞ |x_n|^2 converges to a finite number. The inner product between two sequences x and y is defined by ⟨y,x⟩ = ∑_{n=1}^∞ y_n* x_n, where the complex conjugate appears on the first factor. Holder’s inequality ensures this sum converges and produces a finite complex number, so the inner product is well-defined. With that inner product, ℓ^2 becomes a Hilbert space.

The construction generalizes further through measure theory. Given a measure space (Ω, A, μ), one considers measurable functions f: Ω → ℂ that are square integrable, meaning the integral ∫_Ω |f|^2 dμ is finite. This yields a space of functions, but a subtle issue arises: the quantity ∫_Ω |f|^2 dμ can be zero even when f is not the zero function (for instance, if f differs from zero only on a set of measure zero). Since norms in a vector space must be zero only for the zero vector, the fix is to quotient out those “zero-norm” functions.

The resulting space, denoted L^2 (with the quotient indicated by the non-curved L in the transcript), consists of equivalence classes of square-integrable functions where two functions are treated as the same if their difference has norm zero—equivalently, they agree except on sets of measure zero. On these equivalence classes, the norm becomes well-defined and the space supports a genuine inner product: for equivalence classes represented by G and F, ⟨G,F⟩ = ∫_Ω G* F dμ, again using complex conjugation on the first argument. With this inner product and completeness, L^2 becomes a Hilbert space.

A key takeaway is that ℓ^2 appears as the special case of this framework when Ω is ℕ with counting measure, while the familiar L^2(ℝ) arises when Ω is the real line with Lebesgue measure. Once these examples are in place, the next step is to use the inner-product geometry they provide—especially orthogonality—in more abstract Hilbert spaces.

Cornell Notes

Hilbert spaces are complete inner-product spaces over ℝ or ℂ. The inner product must be positive definite, conjugate symmetric, and linear in the second argument; the norm is then defined by ||x|| = sqrt(⟨x,x⟩). Finite-dimensional spaces like ℝ^n and ℂ^n are automatically complete once an inner product is chosen. In infinite dimensions, ℓ^2 consists of sequences (x_n) with ∑|x_n|^2 < ∞, with inner product ⟨y,x⟩ = ∑ y_n* x_n. A broader construction uses a measure space (Ω, A, μ): square-integrable functions form a candidate L^2, but functions that differ only on measure-zero sets must be identified via a quotient so the norm is truly positive definite; the inner product becomes ⟨G,F⟩ = ∫ G*F dμ.

Why does completeness matter for turning an inner-product space into a Hilbert space?

Completeness requires that every Cauchy sequence (with respect to the norm induced by the inner product) converges to an element inside the space. Without completeness, limits of “well-behaved” sequences might fall outside the space, breaking the geometry needed for analysis. The transcript emphasizes that a Hilbert space is a special Banach space: it is complete under the norm derived from the inner product.

How is the inner product on ℓ^2 defined, and what guarantees it is finite?

For sequences x and y in ℓ^2, the inner product is ⟨y,x⟩ = ∑_{n=1}^∞ y_n* x_n, where * denotes complex conjugation. The sum converges to a finite complex number because Holder’s inequality applies to square-summable sequences, ensuring the inner product is well-defined.

What goes wrong if one tries to define L^2 using only square-integrable functions directly?

The integral ∫_Ω |f|^2 dμ can equal 0 even when f is not the zero function, such as when f is nonzero only on a set of measure zero. That would violate the norm axiom that ||f|| = 0 implies f is the zero vector. The transcript resolves this by quotienting out all functions with zero norm.

What exactly is the quotient construction for L^2, and how does it fix the norm?

Let N be the set of measurable functions f with ∫_Ω |f|^2 dμ = 0. Two functions are treated as equivalent if their difference lies in N, meaning they agree except on measure-zero sets. Elements of L^2 are equivalence classes, and the norm is defined using the original integral on any representative; it becomes well-defined because all zero-norm functions collapse into the same class as the zero function.

How is the inner product on L^2 defined once equivalence classes are used?