How big is a function

Mathematics · Calculus & Analysis · ages 22-24

Name ______________________ Date ____________

Indicator of the interval from 1 to 4. L1 norm equals length. What is it?
Answer: ______________
Which inequality bounds the integral of a product?
- Reversed triangle
- Equality always
- No bound exists
- Holder with product of norms
Which inequality bounds the integral of a product of two functions?
- Minkowski, by the sum of the two norms
- The triangle inequality applied pointwise
- Holder, by the product of the two norms
Lp treats functions that agree almost everywhere as the same element, because a nonzero difference on a null set has norm zero.
Circle one: True False
Lp is a Hilbert space only when p equals 2, because only then does the norm come from an inner product.
Circle one: True False
Lp is Hilbert only when p equals 2 because only then does the norm come from an inner product. Is this correct?
Circle one: True False
Why must almost everywhere equal functions be identified?
- To make spaces smaller
- Otherwise a nonzero function vanishing outside a null set would have zero norm, breaking definiteness
- No reason
- To avoid completeness
Why must almost everywhere equal functions be identified?
- To keep the space small and tidy
- Otherwise a nonzero function vanishing outside a null set would have zero norm
- To avoid completeness arguments
What makes the exponent 2 special among Lp spaces?
- The norm comes from an inner product, bringing orthogonality and the parallelogram law
- The space fails to be complete
- Every function in it is automatically bounded
Function doubles its input on the interval from 0 to 3. Essential supremum is what?
Answer: ______________

Answer key

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

How big is a function W1-mt_Qd8jU26qgE-s1

3 · Length 4 minus 1 equals 3.
Holder with product of norms · Holder with product of norms is correct, giving integral of absolute product at most product of norms.
Holder, by the product of the two norms · Holder is the product bound, while Minkowski governs sums.
True · A difference confined to a null set integrates to 0, so definiteness demands the identification.
True · The parallelogram law holds for the norm only at exponent 2.
True · Parallelogram law holds only at exponent 2.
Otherwise a nonzero function vanishing outside a null set would have zero norm, breaking definiteness · Otherwise a nonzero function vanishing outside a null set would have zero norm, breaking definiteness is the reason, while the others are false.
Otherwise a nonzero function vanishing outside a null set would have zero norm · A nonzero function with zero norm would break definiteness, and only the identification prevents that.
The norm comes from an inner product, bringing orthogonality and the parallelogram law · The inner product structure exists only at exponent 2.
6 · Maximum of 2x on [0, 3] is 6.