Plato argues that space is a consequence, a byproduct of “substance” (Parmenides 145b3-e6): space depends on “substance”.

Immediately afterward, he advances the argument that substance not only generates space but that the space that substance generates is a nested manifold (Parmenides 148d5-149d7).

Subsequently in the Timaeus dialogue he attempts to elaborate the character of this substance-dependent space by employing the following descriptive expressions: imprint-bearer, container, winnowing basket, receptacle, all-recipient, and nurse, nurturer, mother, space, and seat.

(What Plato means by being, substance, the one, or eidos (words that Plato uses interchangeably as signifiers for his usually misunderstood ontological order of incorporeal causes) is another controversial issue to be discussed separately).

Plato’s ontological generation of space is diametrically opposed to the logical construction of the mathematical concept of Hilbert space whose validity depends not on any ontological facts but on the logical consistency of its own a priori axiomatic structure.

It is remarkable that the employment of Hilbert space mathematics by quantum physics appears to have led to certain (still untested) descriptions of the physical world that are more consistent with Plato’s view of reality than they are with Immanuel Kant’s Weltanschauung on which Hilbert’s mathematical achievement is based (here, here, here, here and here). In other words, the mathematical-physics consequences of Hilbert’s mathematics vindicate the ontological suppositions of Plato and not the critical-rational view of Kant, even though Hilbert’s mathematics depend on Kant’s critical rationalism and not on Plato’s ontological realism.

You can read here the relevant passages from Plato in the original Greek side-by-side with my English translations.