Definition:Dimension (Hilbert Space)
This page is about the dimension of a Hilbert space. For other uses, see Definition:Dimension.
Definition
Let $H$ be a Hilbert space, and let $E$ be a basis of $H$.
Then the dimension $\dim H$ of $H$ is defined as $\left\vert{E}\right\vert$, the cardinality of $E$.
Note
It is not obvious that this does not depend on the particular choice of $E$.
This is, however, the case, as proved in Dimension of Hilbert Space is Well-Defined.
In particular: link to other defs of dimension seems appropriate.
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Sources
- John B. Conway: A Course in Functional Analysis (1990)... (previous)... (next) $I.4.15$
