Semi-Hilbert space

In functional analysis, a semi-Hilbert space is a complex or real vector space equipped with a positive semidefinite sesquilinear form whose associated quotient space is a Hilbert space. Semi-Hilbert spaces generalize ordinary Hilbert spaces by allowing nonzero vectors to have zero length.

Unlike the inner product of a Hilbert space, the form defining a semi-Hilbert space need not be positive definite. Consequently, it induces a seminorm rather than necessarily a norm. The degeneracy of this seminorm is removed by taking the quotient with respect to its kernel.

Formal definition

Let $V$ be a vector space over $𝕂$ , where $𝕂 = ℝ$ or $ℂ$ .

Suppose that

⟨ \cdot, \cdot ⟩ : V \times V ⟶ 𝕂

is a positive semidefinite symmetric bilinear form in the real case, or a positive semidefinite Hermitian sesquilinear form in the complex case.

Using the convention that the form is linear in its first argument, it satisfies the following properties.

Sesquilinearity

For all $x, y, z \in V$ and $a, b \in 𝕂$ ,

⟨ a x + b y, z ⟩ = a ⟨ x, z ⟩ + b ⟨ y, z ⟩ .

In the complex case, conjugate linearity in the second argument gives

⟨ x, a y + b z ⟩ = \overline{a} ⟨ x, y ⟩ + \overline{b} ⟨ x, z ⟩ .

Hermitian symmetry

For every $x, y \in V$ ,

⟨ x, y ⟩ = \overline{⟨ y, x ⟩} .

Positive semidefiniteness

For every $x \in V$ ,

⟨ x, x ⟩ \geq 0.

Unlike a Hilbert space, the condition

⟨ x, x ⟩ = 0 ⟹ x = 0

is not required.

Define the associated seminorm by

p (x) = \sqrt{⟨ x, x ⟩} .

Its kernel is the vector subspace

N = \ker p = {x \in V : ⟨ x, x ⟩ = 0} .

The quotient vector space

V / N

inherits a positive definite inner product given by

⟨ x + N, y + N ⟩_{V / N} = ⟨ x, y ⟩_{V} .

A semi-Hilbert space is obtained when this quotient is complete with respect to its induced norm.

Equivalently,

(V, ⟨ \cdot, \cdot ⟩) is semi-Hilbert ⟺ V / N is Hilbert .

Fundamental properties

Cauchy–Schwarz inequality

Every positive semidefinite Hermitian form satisfies the Cauchy–Schwarz inequality

| ⟨ x, y ⟩ |^{2} \leq ⟨ x, x ⟩ ⟨ y, y ⟩ .

Proof.

Suppose first that $⟨ y, y ⟩ > 0$ .

For every scalar $λ \in 𝕂$ , positive semidefiniteness implies

0 \leq ⟨ x - λ y, x - λ y ⟩ .

Expanding,

0 \leq ⟨ x, x ⟩ - \overline{λ} ⟨ x, y ⟩ - λ ⟨ y, x ⟩ + | λ |^{2} ⟨ y, y ⟩ .

Choose

λ = \frac{⟨ x, y ⟩}{⟨ y, y ⟩} .

Substitution yields

0 \leq ⟨ x, x ⟩ - \frac{| ⟨ x, y ⟩ |^{2}}{⟨ y, y ⟩} .

Therefore,

| ⟨ x, y ⟩ |^{2} \leq ⟨ x, x ⟩ ⟨ y, y ⟩ .

If $⟨ y, y ⟩ = 0$ , positivity of

⟨ x + t y, x + t y ⟩

for arbitrary real or complex scalars $t$ implies that $⟨ x, y ⟩ = 0$ . The inequality follows in this case as well.

Characterization of the null space

Proposition. The kernel of the induced seminorm coincides with the radical of the positive semidefinite form:

N = {x \in V : ⟨ x, y ⟩ = 0 for every y \in V} .

Proof.

If $x \in N$ , then

⟨ x, x ⟩ = 0.

By the Cauchy–Schwarz inequality,

| ⟨ x, y ⟩ |^{2} \leq ⟨ x, x ⟩ ⟨ y, y ⟩ = 0.

Hence

⟨ x, y ⟩ = 0

for every $y \in V$ .

Conversely, if $⟨ x, y ⟩ = 0$ for every $y \in V$ , choosing $y = x$ gives

⟨ x, x ⟩ = 0.

Thus $x \in N$ , proving the equality.

Quotient Hilbert space

Theorem. Let $V$ be a vector space equipped with a positive semidefinite Hermitian form, and let $N$ be its null space. Then $V / N$ is an inner product space. It is a Hilbert space if and only if it is complete in the induced norm.

Proof.

Define

⟨ [x], [y] ⟩ = ⟨ x, y ⟩,

where $[x] = x + N$ denotes the equivalence class of $x$ .

To establish that this definition is independent of the representatives, suppose

x^{'} = x + n_{1}, y^{'} = y + n_{2},

with $n_{1}, n_{2} \in N$ .

Then

\begin{matrix} ⟨ x^{'}, y^{'} ⟩ & = ⟨ x + n_{1}, y + n_{2} ⟩ \\ = ⟨ x, y ⟩ + ⟨ n_{1}, y ⟩ + ⟨ x, n_{2} ⟩ + ⟨ n_{1}, n_{2} ⟩ \\ = ⟨ x, y ⟩ . \end{matrix}

The final equality follows from the characterization of the null space.

Therefore the quotient form is well-defined.

Furthermore,

⟨ [x], [x] ⟩ = 0 ⟺ x \in N ⟺ [x] = 0.

Thus the induced form is positive definite.

Consequently, $V / N$ is an inner product space, and it is a Hilbert space precisely when it is complete.

Completeness

Completeness of the quotient is an essential part of the definition.

A positive semidefinite form alone does not guarantee that the quotient is complete.

For example, let

V = C_{c}^{\infty} (ℝ),

the vector space of smooth compactly supported functions on the real line.

Equip it with the positive definite form

⟨ f, g ⟩ = \int_{ℝ} f (x) \overline{g (x)} d x .

Its null space is trivial:

N = {0} .

However, $V$ is not complete in the induced $L^{2}$ norm.

Its completion is

L^{2} (ℝ) .

Thus $V$ , with this form, is a pre-Hilbert space but not a semi-Hilbert space under the quotient-completeness definition.

More generally, every positive semidefinite Hermitian form determines a Hilbert space after quotienting by its null space and subsequently taking the metric completion.

Examples

Degenerate Euclidean space

Consider the vector space

V = ℝ^{3}

with the bilinear form

⟨ x, y ⟩ = x_{1} y_{1} + x_{2} y_{2} .

The associated seminorm is

p (x) = \sqrt{x_{1}^{2} + x_{2}^{2}} .

The null space is

N = {(0, 0, t) : t \in ℝ} .

Hence

V / N ≅ ℝ^{2} .

Since finite-dimensional inner product spaces are complete, the quotient is a Hilbert space.

Therefore $V$ is a semi-Hilbert space.

Sobolev-type seminorm

Let

V = H^{1} (0, 1),

the first-order Sobolev space on the interval $(0, 1)$ .

Define

⟨ f, g ⟩_{D} = \int_{0}^{1} f^{'} (x) \overline{g^{'} (x)} d x .

The induced seminorm is

p (f) = {(\int_{0}^{1} | f^{'} (x) |^{2} d x)}^{1 / 2} .

Its null space consists precisely of the almost-everywhere constant functions:

N = {f \in H^{1} (0, 1) : f^{'} = 0} .

Consequently,

H^{1} (0, 1) / N

is an inner product space with norm determined by the weak derivative.

By the Poincaré inequality, this quotient is complete.

It is therefore a Hilbert space, making $H^{1} (0, 1)$ a semi-Hilbert space under the derivative form.

Relationship to Hilbert spaces

Every Hilbert space is a semi-Hilbert space with trivial null space.

Conversely, a semi-Hilbert space is a Hilbert space under its defining form if and only if

N = {0} .

The canonical quotient mapping

q : V ⟶ V / N

satisfies

‖ q (x) ‖_{V / N} = p (x) .

Thus the quotient identifies vectors whose difference has zero seminorm.

In particular,

q (x) = q (y) ⟺ p (x - y) = 0.

This construction allows positive semidefinite forms to be studied using the ordinary theory of Hilbert spaces.

norm.

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