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When Cauchy sequences find their limit

Mathematics · Calculus & Analysis · ages 22-24

Name ______________________ Date ____________

For the vector (3, -7, 5), what is the max norm, the largest absolute value of an entry?
Answer: ______________
Which proposed length on R2 fails the triangle inequality?
- The sum |x| + |y| of the two absolute values
- The larger of the two numbers |x| and |y|, the max norm
- The Euclidean length, the square root of x squared plus y squared
- The square of the sum of the square roots of |x| and |y|
Which properties must a length function satisfy to be a norm?
- Positivity alone, since lengths can never be negative
- Linearity, so the length of a sum equals the sum of lengths
- Definiteness, homogeneity, and the triangle inequality
The polynomials with the sup norm on the interval from 0 to 1 form an incomplete space, since the exponential is a missing limit.
Circle one: True False
All norms on a finite dimensional space are equivalent, while in infinite dimensions this can fail.
Circle one: True False
On R2, the max norm is at most the Euclidean norm, and the Euclidean norm is at most the square root of 2 times the max norm. Is this true?
Circle one: True False
With the sup norm on [0, 1], in which space do the Taylor partial sums of e to the x have a limit inside the space?
- The polynomials on [0, 1]
- The continuous functions on [0, 1]
- The polynomials of degree at most 10
- The constant functions on [0, 1]
A sequence in R2 is Cauchy in the max norm M. Since the Euclidean norm E satisfies E <= sqrt(2) M, must the sequence also be Euclidean-norm Cauchy?
- No, because the norms give different numerical lengths
- Yes, by the fixed bound E <= sqrt(2) M
- Only if every term has Euclidean norm at most 1
Why are all norms on a finite-dimensional space equivalent?
- The unit sphere is compact, so each norm attains positive bounds
- Every finite dimensional space is incomplete, so norms collapse
- Every norm on a finite dimensional space equals the Euclidean norm
p_n is the Taylor partial sum of e to the x up to degree n. On [0, 1] with the sup norm, what is the norm of p_2 minus p_1?
Answer: ______________

Answer key

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When Cauchy sequences find their limit W1-mt_y_jueKnovX-s1

7 · The absolute values are 3, 7 and 5, and the largest is 7.
The square of the sum of the square roots of |x| and |y| · Take (1, 0) and (0, 1): each has length 1, but their sum (1, 1) gets length (1 + 1) squared = 4, which exceeds 1 + 1.
Definiteness, homogeneity, and the triangle inequality · A norm must satisfy all three axioms.
True · Taylor partial sums are Cauchy yet converge outside the space.
True · Compactness of the sphere gives equivalence, and polynomials give a counterexample beyond finite dimensions.
True · Write M = max(|x|, |y|) and E = sqrt(x^2+y^2). The squares satisfy M^2 <= E^2 <= 2M^2, giving both stated bounds.
The continuous functions on [0, 1] · The continuous functions on [0, 1] with the sup norm are complete and contain e to the x. The other three spaces do not contain it.
Yes, by the fixed bound E <= sqrt(2) M · Apply the bound to differences of sequence terms. Max-norm distances tending to zero force Euclidean distances to tend to zero too.
The unit sphere is compact, so each norm attains positive bounds · On the compact Euclidean unit sphere, another norm has a positive minimum and finite maximum. Scaling these bounds compares the norms; it does not make their values equal.
0.5 · p_2 minus p_1 is x squared over 2, whose largest value on [0, 1] is 0.5. Later gaps shrink like 1 over n factorial.