Baire Category and the Uniform Boundedness Principle · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Pointwise bounds become one uniform bound

Mathematics · Calculus & Analysis · ages 23-24
Name ______________________   Date ____________

  1. Operators T_n satisfy ||T_n|| <= 5 for all n, and ||x|| = 2. What is the largest that sup over n of ||T_n x|| can be?

    Answer: ______________

  2. A student says the rationals form a meagre set in the real line. Is that right?

    Circle one:   True   False

  3. Nested closed intervals start with length 1 and each is half as long as the previous. What is the length of the fifth interval?

    Answer: ______________

  4. What does the category theorem forbid in a complete space?

    • Any dense subset
    • Any convergent sequence
    • A countable union of nowhere dense sets filling it

  5. What does the open mapping theorem add to this picture?

    • Pointwise bounds
    • Surjective bounded operators send open sets to open sets
    • A new definition of limit

  6. Which sequence shows the rationals are not complete?

    • The sequence 1/n
    • Constant sequences
    • Decimal approximations of sqrt(2)
    • The alternating sequence (-1)^n

  7. A surjective bounded linear operator between Banach spaces sends open sets to open sets. A student says this always holds. Is that right?

    Circle one:   True   False

  8. The Baire argument traps a point using nested closed balls with shrinking radii. Which hypothesis makes that trap work?

    • Separability of the space
    • Finite dimensionality
    • Compactness of the unit ball
    • Completeness of the domain

  9. A student claims uniform bounds with no completeness hypothesis. What breaks?

    • Linearity fails
    • Nothing breaks
    • The nested-balls point may fall through a hole

  10. The closed graph theorem hands you boundedness from a closed graph.

    Circle one:   True   False

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Pointwise bounds become one uniform bound W1-mt_11Rd73trAX-s1

  1. 10 · Uniform boundedness caps every operator norm by 5, and ||T_n x|| <= 5 times 2 = 10.
  2. True · Each singleton is nowhere dense and the rationals are countable, so Q is a countable union of nowhere dense sets.
  3. 0.0625 · Halving four times from length 1 leaves (1/2)^4 = 0.0625.
  4. A countable union of nowhere dense sets filling it · Thin pieces cannot stack up to the whole complete room.
  5. Surjective bounded operators send open sets to open sets · It extends the completeness argument to images of sets.
  6. Decimal approximations of sqrt(2) · Those decimals are Cauchy but their only possible limit sqrt(2) is missing from Q, so decimal approximations of sqrt(2) witness incompleteness.
  7. True · Surjective bounded operators between Banach spaces are open maps; that is the open mapping theorem.
  8. Completeness of the domain · Baire needs a complete space to trap a point in nested balls, so completeness of the domain is the load-bearing hypothesis.
  9. The nested-balls point may fall through a hole · Holes swallow the limit the argument needs.
  10. True · It is the sibling extension of the same argument.
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