Let $X$ be an inner product space.

An orthonormal subset $B \subseteq X$ is called:

(i) a maximal orthonormal set if there is no other orthonormal subset of $X$ that contains $B$

(ii) an orthonormaml basis if $\text{span}(B)$ is dense in $X$

Consider the following statements about $X$.

(a) $X$ is complete

(b) every maximal orthonormal subset of $X$ is an orthonormal basis for $X$

Note 1. The converse of (b) (i.e., every orthonormal basis for $X$ is a maximal orthonormal subset of $X$) is always true and easy to prove.

Note 2. The proof that (a) implies (b) is in essentially every textbook that covers Hilbert spaces.

Note 3. If $X$ is separable, then (b) implies (a). See:

• Every incomplete inner product space has a maximal but incomplete orthonormal system
• Example of complete orthonormal set in an inner product space whose span is not dense
• Non total orthonormal set in a non Hilbert inner product space

Question. If X is not separable, does (b) imply (a)?

Let $X$ be an inner product space.

An orthonormal subset $B \subseteq X$ is called:

(i) a maximal orthonormal set if there is no other orthonormal subset of $X$ that contains $B$

(ii) an orthonormal basis if $\text{span}(B)$ is dense in $X$

Consider the following statements about $X$.

(a) $X$ is complete

(b) every maximal orthonormal subset of $X$ is an orthonormal basis for $X$

Note 1. The converse of (b) (i.e., every orthonormal basis for $X$ is a maximal orthonormal subset of $X$) is always true and easy to prove.

Note 2. The proof that (a) implies (b) is in essentially every textbook that covers Hilbert spaces.

Note 3. If $X$ is separable, then (b) implies (a). See:

• Every incomplete inner product space has a maximal but incomplete orthonormal system
• Example of complete orthonormal set in an inner product space whose span is not dense
• Non total orthonormal set in a non Hilbert inner product space

Question. If X is not separable, does (b) imply (a)?