Pre-Hilbert space – its Dual and Completion

Posted on 2026/05/29 by arkajad

This post is something of an interlude. I was supposed to be writing about complexification: how to turn real vector spaces into complex ones, and what that means for quantum mechanics.

In the complex setting, the standard formulation of quantum mechanics is well established. But the physical world is real. If complex numbers are “just a tool” to obtain real predictions, then there ought to be a formulation of quantum mechanics that is developed as far as possible purely in the real setting. Only when it becomes genuinely convenient for computation should one pass to complex numbers, and then one should say so explicitly. Textbooks almost never do this. I plan to return to these questions in a future post.

When I started drafting the complexification post, I found myself getting sidetracked by a more basic analytic issue: pre‑Hilbert spaces and their completion to Hilbert spaces (real or complex—this point will not matter). I have never liked the standard construction of the completion via Cauchy sequences and equivalence classes. It feels too abstract and ad hoc; I cannot imagine Nature “using” that construction. There ought to be a more direct and conceptual method.

It seemed natural to suspect that the dual of a pre‑Hilbert space should provide exactly what we need: it should be isomorphic, as a Banach space, to the usual completion obtained from Cauchy sequences. However, when I started looking in the literature, I could not find this stated explicitly in the form I wanted.

Eventually I did find almost exactly the statement I was after—but not in English. The French Wikipedia article on pre‑Hilbert spaces contains the following sentence:

L’image canonique φ(E) de E dans son dual est dense. Le dual s’identifie donc à l’espace de Hilbert complété de l’espace préhilbertien φ(E).

Unfortunately, no proof is given there.

At that point I turned to AI for help. I first asked Perplexity whether the French statement is correct. It responded cautiously, beginning with: “Key point: that French sentence is subtly misstated in the generality of ‘any pre‑Hilbert space’,” and then continued without ever committing to a clear, unambiguous assertion. So I tried Grok. Grok agreed with the French Wikipedia formulation and supplied a “Sketch of a proof”. I then presented Grok’s argument back to Perplexity, which replied:

“I do have a concrete objection to Grok’s reasoning as written … The proof uses an unjustified extension step and quietly reverses the direction of the standard extension–restriction correspondence between …”

and so on.

At that point I decided to stop arguing with AIs and to write down my own proof of the statement I believed to be true. It took some time to make the argument logically precise and internally clean—clean enough that Perplexity eventually accepted it. This post is the result.

Note: You can find an illustrated, colored version of the above preamble in my Substack post.
The pdf is availableon ResearchGate  here.

  1. Introduction.
    We start with recalling two well–known results.

Theorem 1 [Extension of bounded linear maps].
Let V be a normed space, W\subseteq V a dense linear subspace, Z a Banach space and A:W\rightarrow Z a bounded linear map. Then there is a unique bounded linear map \hat{A}:V\rightarrow Z with A=\hat{A}_{W}. It satisfies \Vert\hat{A}\Vert=\Vert A\Vert.
Proof.  See Ref [1], Lemma 3.12, p. 22], adapted to the real case.

Theorem 2 [Riesz Representation Theorem]. If H is a (real) Hilbert space and \phi\in H^* then there is a unique y\in H such that \phi=\phi_y=(\cdot,y). The map H\rightarrow H^*,\, y\mapsto \phi_y is a linear isometric bijection.

Proof. See Ref. [1], Theorem 5.34, p. 45], adapted to the real case.

2. Dual of a dense subspace of a Hilbert space

Let (\hat{H},(\cdot,\cdot)) be a real Hilbert space. With the norm \Vert\cdot\Vert defined by

(1)   \begin{equation*} \Vert x\Vert^2=(x,x),\end{equation*}

\hat{H} is a Banach space.

Let D be a dense subspace of \hat{H}. Then, with the norm inherited from \hat{H}, D is a normed space. We denote by D^* its dual vector space. Equipped with the “sup” norm D^* is a Banach space.

We define the linear map \Phi: \hat{H}\rightarrow D^* by

(2)   \begin{equation*} \Phi(x)(y)=(x,y),\quad x\in \hat{H},\, y\in D.\end{equation*}

Lemma 1.  The map \Phi is an isomorphism of Banach spaces \hat{H} and D^*.
Remark.  The term “isomorphism of Banach spaces” includes the fact that \Phi is an isometry.
Proof (of Lemma 1). We recall that we work over the reals, so the scalar product (x,y) is symmetric and linear in x and in y. Thus the map \Phi(x):y\mapsto (x,y) is linear, so \Phi(x) is a linear functional on D. Moreover, \Phi(x) is a continuous linear functional on D, since the scalar product is continuous (in both arguments). The map x\mapsto \Phi(x) from \hat{H} to D^* is also linear in x .

The map \Phi is an injection. Indeed, assuming \Phi(x)=0 we obtain (x,y)=0, for all y\in D, therefore x=0, because D is dense in \hat{H}. QED

We will now show that \Phi is surjective. Let f\in D^*, and let \hat{f} be the unique element of \hat{H}^* extending f, as in Theorem 1 above, moreover \Vert \hat{f}\Vert=\Vert f\Vert. Since \hat{f}\in \hat{H}^*, by the Riesz Representation Theorem (see Theorem 2 above) there is a unique x\in \hat{H} such that \hat{f}(y)=(x,y) for all y\in \hat{H}. Moreover \Vert \hat{f}\Vert=\Vert x \Vert. It follows, in particular, that

(3)   \begin{equation*} \hat{f}(y)= f(y)= (x,y)\end{equation*}

for all y\in D. Comparing with (2) we have that f=\Phi(x). Moreover, \Vert \Phi(x)\Vert=\Vert f\Vert=\Vert\hat{f}\Vert=\Vert x\Vert. QED

Completion of a pre–Hilbert space

We first establish the definitions and the notation. Let H be a real pre–Hilbert space, \hat{H} the normed space which is the completion of H. The scalar product (\cdot,\cdot) on H extends by continuity to a non–degenerate positive definite symmetric form on \hat{H}\times\hat{H}, which defines on \hat{H} a structure of a Hilbert space. We call this space the completion of the pre–Hilbert space H. (cf. Proposition 4, Ch. V.6 in Ref. [2]).

Proposition 1. There exist an isomorphism of Banach spaces \hat{H} and H^*.

Proof. With the notation as above H is a dense subspace of \hat{H}. Therefore we can apply Lemma 1, taking D=H.

Corollary 1.  With the notation as above \Phi(H) is dense in H^*.

Proof. We know that \Phi is a linear isometric injection H\rightarrow H^*, and that H is dense in \hat{H}. It follows that H is dense in \hat{H}. Therefore \Phi(H) is dense in H^*.

Refrences.

[1] Müger, M., “Introduction to Functional Analysis.”
[2] Bourbaki, N., “Topological Vector Spaces Chapters 1–5”, Springer 2003.

 

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One Response to Pre-Hilbert space – its Dual and Completion

  1. John G says:
    2026/05/30 at 01:43

    Your recent comments here and elsewhere seem related to the idea of a real Bott periodicity in a loop space in a phase space in an EEQT Born reciprocity central sector in a Krein space. I still think in the Clifford with enhancements sense that Cl(8,C) is your pre-Hilbert space and you take 8-dim slices of the Cl(8,C) infinite tensor product Krein space for worldline histories.

    What made you think to ask about Bott periodicity? I only knew about it for Clifford algebra and even now that only loosely fits for me with the loop space version I read about on Wikipedia as in a 3-1-3 structure (like music or colors or octonion imaginaries are loose analogies).

    I think I now see why you never really had the need to look at radically different Planck scale math. Even 7th density could fit with the other six in the same structures you work with. It’s been recently odd in a nice way to think of 7th density in the same universe state as me.

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