Math Problems Vault

Problem 13

If $\mathbf{M}$ is a finite-dimensional linear manifold in a Hilbert space $\mathbf{H}$ , and if $\mathbf{N}$ is a subspace (a closed linear manifold) in $\mathbf{H}$ , then the vector sum $\mathbf{M} + \mathbf{N}$ is necessarily closed (and is therefore equal to the span $\mathbf{M} \vee \mathbf{N}$ ).

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Problem 14

For which cardinal numbers $m$ is the lattice of subspaces of a Hilbert space of dimension $m$ modular? distributive?

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Remark.

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The collection of all subspaces of a Hilbert space is a lattice. This means that the collection is partially ordered (by inclusion), and that any two elements $\mathbf{M}$ and $\mathbf{N}$ of it have a least upper bound or supremum (namely the span $\mathbf{M} \vee \mathbf{N}$ ) and a greatest lower bound or infimum (namely the intersection $\mathbf{M} \cap \mathbf{N}$ ). A lattice is called distributive if (in the notation appropriate to subspaces)

\mathbf{L} \cap (\mathbf{M} \vee \mathbf{N}) = (\mathbf{L} \cap \mathbf{M}) \vee (\mathbf{L} \cap \mathbf{N})

identically in $\mathbf{L}$ , $\mathbf{M}$ , and $\mathbf{N}$ .

There is a weakening of this distributivity condition, called modularity; a lattice is called modular if the distributive law, as written above, holds at least when $\mathbf{N} \subset \mathbf{L}$ . In that case, of course, $\mathbf{L} \cap \mathbf{N} = \mathbf{N}$ , and the identity becomes

\mathbf{L} \cap (\mathbf{M} \vee \mathbf{N}) = (\mathbf{L} \cap \mathbf{M}) \vee \mathbf{N}

(with the proviso $\mathbf{N} \subset \mathbf{L}$ still in force).

Problem 15

For subspaces $\mathbf{M}$ and $\mathbf{N}$ of a Hilbert space, the vector sum $\mathbf{M} + \mathbf{N}$ is closed if and only if the modular equation

\mathbf{L} \cap (\mathbf{M} \vee \mathbf{N}) = (\mathbf{L} \cap \mathbf{M}) \vee \mathbf{N}

is true whenever $\mathbf{N} \subset \mathbf{L}$ .

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Problem 16

A Hilbert space is locally compact if and only if it is finite-dimensional.

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Problem 17

A Hilbert space $\mathbf{H}$ is separable if and only if $\dim \mathbf{H} \leq \aleph_0$ .

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Problem 18

For each measure in an infinite-dimensional Hilbert space, the measure of every non-empty ball is infinite.

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Remark.

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Here "measure" refers to a countably additive set function $\mu$ defined on (at least) the collection of all Borel sets (the $\sigma$ -field generated by the open sets), so that $0 \leq \mu (M) \leq \infty$ for all Borel sets $M$ , such that every non-empty open set has positive measure and $\mu$ is invariant under translation (i.e., $\mu (f+M) = \mu (M)$ for every vector $f$ and for every Borel set $M$ ).