Hilbert space has always an orthonormal Schauder basis + what is meant by the dimension of a Hilbert space?

Question

I realized that I have become a bit rusty when it comes to the definition of a basis and have the following question:

Usually, a basis in a Hilbert space, denoted as $H$, is defined as an orthonormal set that is also a Schauder basis. In other words, it's a set $B := (e_i | i \in I) \subset H$, where $I$ may have any cardinality, and $\langle e_i, e_j \rangle = \delta_{ij}$ for all $i$ and $j$, such that $\mathrm{span} (B) := \left(\sum_{j \in J} c_j e_j \bigg| J \subset I, |J| < \infty, \text{ and } c_j \in \mathbb{F} \text{ for all } j \in J \right)$ is dense in $H$.

Now, my question would be: Is the dimension of a Hilbert space the number of elements in $B$? (Is this then well-defined because if there is another such basis, is the number then automatically the same etc.)

And does such a basis always exist for a Hilbert space? (Maybe this can be done with Zorn's Lemma?)

Perhaps some seasoned mathematicians can provide me with a quick answer to these questions to help me gain more clarity.

See here for a proof of existence of orthonormal bases via Zorn's lemma — Rhys Steele, Sep 17 '23 at 20:49
BTW, you should call this the orthogonal dimension. There are different notions of dimension. — geetha290krm, Sep 17 '23 at 23:14
All total orthonormal sets in a Hilbert space have the same cardinality. It is called the Hilbert dimension or orthogonal dimension. And two Hilbert spaces over the same field are isomorphic if and only if they have the same Hilbert dimension. — Ricky, Sep 18 '23 at 01:25
@Ricky Thank you very much. Do you know a reference where I can read about "Hilbert dimensions" or "orthogonal dimensions"? — MackeyTopology, Sep 18 '23 at 01:27
The Wikipedia link has all the facts listed. For proof just consult any functional analysis book, I first learned about them in Kreyszig's Introductory Functional Analysis. — Ricky, Sep 18 '23 at 01:30
@Ricky, would you consider writing your comments into an answer to help remove this from the Unanswered queue? It appears that you have satisfied the OP's request to their satisfaction. — Eric Nathan Stucky, Sep 18 '23 at 05:18

score 4 · Accepted Answer · answered Sep 18 '23 at 05:50

First of all, the dimension of a Hilbert space can mean Hamel dimension or Hilbert dimension, which are in general different (Hilbert dimension is no greater than Hamel dimension).

The Hilbert dimension (or orthogonal dimension) of a Hilbert space is defined as the cardinality of a total orthonormal set (orthonormal basis) in that space. It is well-defined because all total orthonormal sets in a Hilbert space have the same cardinality. And two Hilbert spaces over the same field are isomorphic if and only if they have the same Hilbert dimension.

The Hamel dimension of any vector space is the cardinality of a basis of the space over its base field. It is also well-defined because all Hamel bases of a vector space have the same cardinality.

As a result of Zorn's Lemma, every nontrivial vector space has a Hamel basis, and every nontrivial Hilbert space has an orthonormal basis. These two bases coincide if and only if the space is finite dimensional. (I learned about these in Kreysig's Introductory Functional Analysis p212)

Hilbert space has always an orthonormal Schauder basis + what is meant by the dimension of a Hilbert space?

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