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In class, my professor mentioned that a zero dimensional Hilbert space is essentially just a field of scalars. I don't see how this would be, because this sounds like a one dimensional Hilbert space. I haven't dealt with zero dimensional spaces before. Can someone explain? How are zero dimensional spaces useful?

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    See [wikipedia](https://en.wikipedia.org/wiki/Zero-dimensional_space) for references, .i.e., usefulness. – 2017-01-12
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    Your intuition is correct: a field of scalars is a space of dimension $1$. A space of dimension $0$ is nothing but the set $\{ 0 \}$. – 2017-01-12
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    So just a single point? Can it be any point in one-dimensional complex Hilbert space, or does it have to be 0? – 2017-01-12
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    @Dietrich Burde: Since the question involves Hilbert spaces, I suspect radm94 means algebraic dimension and not topological dimension. – 2017-01-12
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    Regarding singleton sets in vector spaces and being zero dimensional, see the google search ["vector space" + "zero dimension"](https://www.google.com/search?q=%22vector+space%22+%22zero+dimension%22) (which includes several related stackexchange questions). – 2017-01-12
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    @DaveL.Renfro Who knows. I think of "Zero-dimensional Rings" with Krull dimension. There is a whole book on such rings. So why not consider interesting "Zero-dimensional Hilbert spaces". – 2017-01-12
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    @DaveL.Renfro I didn't know there were two separate things. I think we were in fact discussing algebraic dimension (which I think deals with the rank-nullity theorem). What would the topological dimension of $\{0\}$ be? – 2017-01-12
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    @radm94 One usually just writes $0$ to be the additive identity, not in reference to any larger space which it might be a part of. If there's only one point, it's certainly the additive identity, hence notated by $0$. (Though, the only subspace of dimension $0$ is always just $\{0\}$) – 2017-01-12
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    The topological dimension of $\{0\}$ (and any finite set of points in a reasonably nice metric space) is zero. There are many varieties of "dimension" in mathematics, although not all will apply in your situation. @Dietrich Burde mentioned Krull dimension, while I'm more familiar with various "fractal dimensions" (Hausdorff dimension, Minkowski dimension, packing dimension, etc.). For normed linear spaces there is the Hamel dimension and the Schauder dimension, and also the various topological dimensions. – 2017-01-12

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