Chapter 3

Weak Topology

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Problem 19

Every weakly closed set is strongly closed, but the converse is not true. Nevertheless every subspace of a Hilbert space (i.e., every strongly closed linear manifold) is weakly closed.

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Remark.
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The metric topology (or norm topology) of a Hilbert space is often called the strong topology. A base for the strong topology is the collection of open balls, i.e., sets of the form {f:∥f−f0∥<ε}\left\{ f : \left\| f - f_0\right\| < \varepsilon \right\}, where f0f_0 (the center) is a vector and ε\varepsilon (the radius) is a positive number.

A subbase (not a base) for the weak topology is the collection of all sets of the form {f:∣⟨f−f0,g0⟩∣<ε}\left\{ f : \left| \left\langle f - f_0, g_0 \right\rangle \right| < \varepsilon \right\}. It follows that a base for the weak topology is the collection of all sets of the form {f:∣⟨f−f0,gi⟩∣<ε,i=1,…,k}\left\{ f : \left| \left\langle f - f_0, g_i \right\rangle \right| < \varepsilon , i = 1, \ldots , k\right\}, where kk is a positive integer, f0,g1,…,gkf_0, g_1, \ldots , g_k are vectors, and ε\varepsilon is a positive number.

A sequence, or a net, {fn}\left\{ f_n\right\} is weakly convergent to ff if and only if ⟨fn,g⟩→⟨f,g⟩\left\langle f_n, g \right\rangle \to \left\langle f, g \right\rangle for each gg.

Problem 20

If fn→ff_n \to f (weak) and ∥fn∥→∥f∥\left\| f_n\right\| \to \left\| f\right\|, then fn→ff_n \to f (strong).

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Remark.
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A sequence, or a net, {fn}\left\{ f_n\right\} is weakly convergent to ff if and only if ⟨fn,g⟩→⟨f,g⟩\left\langle f_n, g \right\rangle \to \left\langle f, g \right\rangle for each gg.

Problem 21

Norm is weakly lower semicontinuous.

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Remark.
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Explicitly: if fn→ff_n \to f (weak), then ∥f∥≤lim inf⁡n∥fn∥\left\| f\right\| \leq \liminf_n \left\| f_n\right\|. Equivalently: for every ε>0\varepsilon > 0, there exists an n0n_0 such that ∥f∥≤∥fn∥+ε\left\| f\right\| \leq \left\| f_n\right\| + \varepsilon whenever n≥n0n \geq n_0.

Problem 22

Is every weakly separable Hilbert space separable?

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Problem 23

The closed unit ball in a Hilbert space is weakly compact.

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Remark.
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The result is sometimes known as the Tychonoff--Alaoglu theorem. It is as hard as it is important. It is very important.

Problem 24

Is the weak topology of the unit ball in a separable Hilbert space metrizable?

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Problem 25

What is the weak closure of the unit sphere (i.e., of the set of all unit vectors)? If a set is weakly dense in a Hilbert space, does it follow that its intersection with the unit ball is weakly dense in the unit ball?

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Problem 26

If the weak topology of the unit ball in a Hilbert space H\mathbf{H} is metrizable, must H\mathbf{H} be separable?

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Problem 27

Find an elementary proof of the principle of uniform boundedness for Hilbert space.

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Remark.
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In this context a proof is "elementary" if it does not use the Baire category theorem.

Problem 28

The weak topology of an infinite-dimensional Hilbert space is not metrizable.

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Problem 29

If ∑n∣αnβn∣<∞\sum_n \left|\alpha_n \beta_n\right| < \infty whenever ∑n∣αn∣2<∞\sum_n \left|\alpha_n\right|^2 < \infty, then

∑n∣βn∣2<∞. \sum _n \left|\beta _n\right|^2 < \infty .
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Problem 30
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(a)

No infinite-dimensional Hilbert space is weakly complete.

(b)

Which Hilbert spaces are sequentially weakly complete?

Remark.
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A sequence {gn}\left\{ g_n\right\} of vectors in a Hilbert space is a weak Cauchy sequence if the numerical sequence (⟨f,gn⟩)\left( \left\langle f, g_n \right\rangle \right) is a Cauchy sequence for each ff in the space. Weak Cauchy nets are defined exactly the same way: just replace "sequence" by "net" throughout. To say of a Hilbert space, or a subset of one, that it is weakly complete means that every weak Cauchy net has a weak limit (in the set under consideration). If the conclusion is known to hold for sequences only, the space is called sequentially weakly complete.