Norms: Tutorial problems
1. Let X be a normed vector space and let x, y ∈ X. Show that
||x|| − ||y|| ≤ ||x − y||.
2. Suppose xn is a Cauchy sequence in a normed vector space X.
Show that ||xn || is a convergent sequence in R.
3. Show that the unit sphere is closed in every normed vector space.
4. Consider the discrete metric on R. Is it induced by a norm?
5. Consider the space X = C[0, 1] and the identity function
I : (X, || · ||∞ ) → (X, || · ||1 ),
I(f (x)) = f (x).
Show that I is Lipschitz continuous, but its inverse is not continuous.
6. Show that ℓp ⊂ ℓq whenever 1 ≤ p < q ≤ ∞.
Norms: Some hints
1. Show that ||x|| ≤ ||y|| + ||x − y|| and ||y|| ≤ ||x|| + ||x − y||.
2. Show that ||xn || is a Cauchy sequence in R. Use the first problem.
3. The norm function f : X → R is continuous in every normed vector
space X. Since {1} is closed in R, its inverse image is closed in X.
4. If a metric d is induced by a norm, then d(2x, 2y) = 2d(x, y).
5. For the first part, show that ||f − g||1 ≤ ||f − g||∞ for all f, g ∈ X.
For the second part, show that d1 (xn , 0) → 0, whereas d∞ (xn , 0) = 1.
P
6. If x ∈ ℓp , then
|xn |p < ∞, so xnP→ 0 by the nthP
term test. This
q
q−p
|xn |p < ∞.
makes xn bounded, say |xn | ≤ R, so
|xn | ≤ R