Functional Analysis 9 | Examples of Inner Products and Hilbert Spaces [dark version]

TL;DR

Hilbert spaces require an inner product plus completeness under the induced norm, not just the ability to define ⟨·,·⟩.

Briefing Cornell Notes

Briefing

Hilbert spaces come from vector spaces equipped with an inner product that makes the induced metric complete—but having an inner product alone is not enough. The transcript walks through several core examples, then zeroes in on why the standard inner product on L2 really works, laying out the three defining inner-product properties (positivity, conjugate symmetry, and linearity) and explaining why those checks are usually straightforward compared with the harder “well-definedness” step.

As a starting point, finite-dimensional Euclidean spaces illustrate the template: R^n or C^n becomes a Hilbert space when equipped with the usual inner product that sums componentwise products, with complex conjugation applied to the first component in the complex case. The transcript then generalizes to infinite dimensions via L2, the space of square-integrable functions. The inner product there is built by integrating the product of one function with the complex conjugate of the other over the domain (the exact formula is presented as an integral over the relevant variable).

A second infinite-dimensional example uses continuous functions on the unit interval [0,1]. Continuous functions form a vector space under pointwise addition and scalar multiplication, and an inner product can be defined using an integral from 0 to 1 of f(t) times the complex conjugate of g(t). This gives the familiar “geometry” of inner products for functions, but it still fails to produce a Hilbert space because completeness breaks down. The key takeaway is that an inner product can exist without the space being complete under the induced norm, so the Hilbert-space requirement is stricter than merely defining an inner product.

The transcript then focuses on L2 and treats the inner product as a map from L2 × L2 to the underlying field. Before checking the inner-product axioms, it notes a technical hurdle: the map must be well-defined, meaning the relevant series/integral expression must converge appropriately. That well-definedness is flagged as a harder issue deferred to a later part of the series.

With that caveat, the three inner-product properties are verified. Positivity is shown by computing ⟨x,x⟩ as a sum/integral of |x_i|^2 (or the function analogue), which is always non-negative, and it equals zero only when every component (or the function) is zero—so the inner product is positive definite. Conjugate symmetry follows by comparing ⟨y,x⟩ with ⟨x,y⟩ and using complex conjugation to reverse the order. Linearity in the second argument is demonstrated by splitting ⟨x, y+z⟩ into ⟨x,y⟩+⟨x,z⟩ and pulling out scalars to show ⟨x,λy⟩ = λ⟨x,y⟩.

Finally, combining these inner-product axioms with the previously established completeness of L2 under its induced norm yields the Hilbert-space conclusion. The transcript closes by emphasizing that once the inner-product properties are checked—and completeness is secured—L2 earns its place as a Hilbert space, enabling the full toolkit of Hilbert-space geometry for infinite-dimensional analysis.

Cornell Notes

Hilbert spaces require more than an inner product: the norm induced by the inner product must make the space complete. The transcript compares three examples—R^n/C^n, L2, and continuous functions on [0,1]. Continuous functions admit an inner product via an integral, but they fail to be Hilbert spaces because completeness does not hold. For L2, the standard integral-based inner product is checked against the three axioms: positivity (⟨x,x⟩ ≥ 0 and equals 0 only for the zero vector), conjugate symmetry (⟨y,x⟩ = conjugate(⟨x,y⟩)), and linearity in the second argument (distributivity and scalar pull-out). Completeness for L2 is then used to conclude it is a Hilbert space.

Why doesn’t the existence of an inner product automatically make a space a Hilbert space?

A Hilbert space is an inner-product space whose induced metric (equivalently, induced norm) is complete. The transcript’s continuous-function example on [0,1] defines an inner product using an integral, but completeness fails, so the space is not Hilbert even though the inner-product axioms hold.

How is positivity of an inner product typically verified in these examples?

Positivity comes from computing ⟨x,x⟩ and showing it becomes a sum/integral of absolute squares. In the complex setting, complex conjugation ensures ⟨x,x⟩ = |x|^2-type expression, which is always non-negative. It equals zero only when every component/function value contributing to the sum/integral is zero, forcing x to be the zero vector.

What does conjugate symmetry mean, and how does complex conjugation enforce it?

Conjugate symmetry requires ⟨y,x⟩ = conjugate(⟨x,y⟩). The transcript shows that swapping the roles of x and y flips the complex conjugation, so the algebra naturally produces the conjugate of the original expression, matching the axiom.

Why is linearity in the second argument checked separately from positivity and symmetry?

Linearity is an independent axiom: it must hold for sums and scalar multiples in the second slot. The transcript demonstrates ⟨x, y+z⟩ = ⟨x,y⟩ + ⟨x,z⟩ by distributing the integral/sum, and ⟨x, λy⟩ = λ⟨x,y⟩ by pulling the scalar out of the expression.

What technical step can be harder than verifying the three inner-product axioms?

The transcript flags “well-definedness” of the inner product map: it must be guaranteed that the expression used to compute ⟨f,g⟩ (often an infinite series or an integral) actually converges/exists for all pairs in the space. The positivity/symmetry/linearity checks are described as comparatively straightforward once the expression is meaningful.

How does completeness combine with the inner-product axioms to yield a Hilbert space?

Once the inner product satisfies positivity, conjugate symmetry, and linearity, it defines a norm. If the space is complete under that norm (the transcript references completeness already established for L2 in an earlier part), then the space qualifies as a Hilbert space.

Review Questions

What completeness property distinguishes a Hilbert space from a general inner-product space?
For complex vector spaces, where does complex conjugation appear in the inner product, and why is it essential?
In verifying an inner product, which step is often more technical: checking the axioms or proving the inner product is well-defined?

Key Points

1
Hilbert spaces require an inner product plus completeness under the induced norm, not just the ability to define ⟨·,·⟩.
2
Continuous functions on [0,1] can be given an integral-based inner product, but they are not Hilbert spaces because completeness fails.
3
The standard inner product on C^n uses componentwise products with complex conjugation on the first component.
4
For L2, the inner product’s positivity follows from expressing ⟨x,x⟩ as a non-negative absolute-square quantity that vanishes only for the zero vector.
5
Conjugate symmetry in complex spaces is enforced by complex conjugation when swapping the arguments of the inner product.
6
Linearity in the second argument is verified by distributing over sums and pulling out scalars from the second slot.
7
Well-definedness (ensuring the inner-product expression exists/converges) can be harder than checking the three inner-product axioms.

Highlights

An inner product can exist without producing a Hilbert space: continuous functions on [0,1] fail completeness even with an integral inner product.

For complex inner products, complex conjugation is the mechanism that makes conjugate symmetry work.

L2 becomes a Hilbert space once the inner-product axioms are satisfied and completeness under the induced norm is in place.