The Hahn-Banach Theorem and the Dual Space · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Measure vectors by testing them everywhere

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

  1. What is the kernel of a nonzero functional?

    • The whole space
    • The hyperplane perpendicular to its arrow
    • Only the zero arrow

  2. A student says the kernel of a continuous linear functional is always closed. Is that right?

    Circle one:   True   False

  3. On R squared with Euclidean norm, f(x, y) = 3x + 4y. What is the dual norm of f?

    Answer: ______________

  4. What does a linear functional do to vectors?

    • Collapses each to a number along a fixed direction
    • Rotates each by ninety degrees
    • Deletes the zero vector

  5. A student says the Hahn-Banach proof needs the axiom of choice. Is that right?

    Circle one:   True   False

  6. The dual of the absolutely summable sequences carries the sup norm. What is that norm of the sequence (1, -2, 0.5, 0, 0, ...)?

    Answer: ______________

  7. How do you separate a point x from a closed subspace M not containing x?

    • A continuous functional that is zero on the subspace but not at the point
    • The zero functional
    • An unbounded functional
    • A compact operator

  8. What is the dual pairing in miniature?

    • A matrix squared
    • A row vector eating a column vector
    • Two columns added together

  9. Every functional on finite-dimensional space is dotting with some arrow. What sets its norm?

    • The dimension count
    • The kernel size
    • The arrow length

  10. When does the geometric Hahn-Banach theorem guarantee strict separation by a hyperplane?

    • Any two disjoint sets whatever their shape
    • Two disjoint nonempty convex sets with one open
    • Any two overlapping convex sets
    • A point and an open set containing it
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Answer key

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

Measure vectors by testing them everywhere W1-mt_pJuHFxkvjS-s1

  1. The hyperplane perpendicular to its arrow · Zero readings sit exactly across from the arrow.
  2. True · Preimages of closed sets under continuous maps are closed, and {0} is closed.
  3. 5 · On Euclidean space the dual norm is the Euclidean norm: sqrt(9 + 16) = 5.
  4. Collapses each to a number along a fixed direction · One fixed arrow sets the measurement for every input.
  5. True · The standard proof well-orders extensions with Zorn's lemma, a choice principle.
  6. 2 · The dual of l1 carries the sup norm, and the largest entry in size is 2.
  7. A continuous functional that is zero on the subspace but not at the point · Hahn-Banach separation produces a continuous functional that is zero on the subspace but not at the point.
  8. A row vector eating a column vector · One-by-n times n-by-one is functional meets vector.
  9. The arrow length · Longer arrow means steeper flattening, hence larger norm.
  10. Two disjoint nonempty convex sets with one open · The geometric Hahn-Banach theorem strictly separates two disjoint nonempty convex sets with one open by a closed hyperplane.
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