- If you have two norms on Fn or more generally on X a finite dimensional vector space (You
can’t escape these. They are encountered as subspaces of Fn for example.) and these are
equivalent norms in the sense that there are scalars δ,Δ not depending on x such that for all
then the open sets defined in terms of these norms are the same.
- Explain carefully why in ℝn,
- Explain carefully why if x is a limit point of a set A with respect to then it is also a limit point
with respect to
1 if the two norms are equivalent as in 2.20. Also show that if xn → x with
, then the same is true with respect to
1 an equivalent norm satisfying
- A vector space X with field of scalars either ℝ or ℂ is called a normed linear space if it has a norm
. That is, the following assumptions are satisfied.
A normed linear space is called a Banach space if, in addition to having a norm, it is
complete. This means that if is a Cauchy sequence, then it must converge. Recall that a
sequence is a Cauchy sequence if for every
ε > 0 there exists N such that if n,m ≥ N,
The space of bounded linear transformations ℒ which are linear transformations mapping
Y consists of the functions L : X → Y such that L is linear,
, called the operator norm is a norm on ℒ and that
Also show that if you have L ∈ℒ and
M ∈ℒ for
X,Y,Z normed linear spaces,
- If you have X a normed linear space and Y is a Banach space, show that ℒ is a Banach space
with respect to the operator norm.
- If X,Y are normed linear spaces, verify that A : X → Y is in ℒ if and only if
A is continuous
at each x ∈ X if and only if A is continuous at 0.
- Verify that Theorem 2.1.17 holds exactly the same if you are in a normed linear space. That is the
dist function is still Lipschitz continuous.
- Suppose X is a Banach space and is a sequence of closed sets in
X such that Bn ⊇ Bn+1 for all
n and no Bn is empty. Also suppose that the diameter of Bn converges to 0. Recall the diameter is
Thus these sets Bn are nested and diam
→ 0. Verify that there is a unique point in the
intersection of all these sets.
- If X is a Banach space, and Y is the span of finitely many vectors in X, show that Y is
- If X is an infinite dimensional Banach space, show that there exists a sequence
n=1∞ such that
≤ 1 but for any m≠n,
≥ 1∕4. Thus in infinite dimensional Banach spaces, closed and
bounded sets are no longer compact as they are in Fn.
- Generalize Lemma 2.2.8 and Lemma 2.2.9 to the case of a Banach space. That is, show that compact
and sequentially compact are the same for a Banach space. Also show that Theorem 2.2.25 holds with
no change for a compact subset of a Banach space.
- In the proof of the fundamental theorem of algebra, explain why there exists z0 such that for p a
polynomial with complex coefficients,
- Explain why a compact set in ℝ has a largest point and a smallest point. Now if f : K → ℝ for K
compact and f continuous, give another proof of the extreme value theorem from using that f is
- A function f : X → ℝ for X a normed linear space is lower semicontinuous if, whenever
xn → x,
It is upper semicontinuous if, whenever xn → x,
Explain why, if K is compact and f is upper semicontinuous then f achieves its maximum
and if K is compact and f is lower semicontinuous, then f achieves its minimum on
- Suppose fn : S → Y where S is a nonempty subset of X a normed linear space and suppose that Y is
a Banach space (complete normed linear space). Generalize the theorem in the chapter to this case:
Let fn : S → Y be bounded functions:supx∈S =
Cn < ∞. Then there exists bounded
f : S → Y such that limn→∞ = 0 if and only if
is uniformly Cauchy. Also show that
BC is a Banach space.
- Show that no interval
⊆ ℝ can be countable. Hint: First show is not countable. You might
do this by noting that every point in this interval can be written as
k=1∞2−kak where ak is either
0 or 1. Let ℱ be ∪nP
. Explain why ℱ is countable. Then let S≡P
why S is uncountable. Let C be all points of the form ∑
k=1m2−kak where ak is 0 or 1. Explain why
C is countable. Let J =
∖ C. Now let θ : S → J be given by θ =
Explain why θ is one to one onto J. If is countable, show there are onto mappings as
showing that S is countable after all.
- Using the above problem as needed, let B be a countable set of real numbers. Say B =
. Explain why g is continuous on ℝ ∖ B and discontinuous on B. Note
that B could be the rational numbers.
For f a continuous function defined on
, extend it to have f =
x > 1 and
x < −1. Consider
This involves elementary calculus and change of variables. Show that pn is a polynomial and that
pn converges uniformly to f on . This is the way Weierstrass originally proved the famous
- In fact the Bernstein polynomials apply for, f having values in a normed linear space and a similar
result will hold. Give such a generalization.
- Consider the following
where f :
n → X a normed linear space. Show
→ 0 as min
Do consider Lemma 2.8.1 first and you may see how to do this.
- Theorem 2.2.41 gave an example of a function which is everywhere continuous and nowhere
differentiable. The first examples of this sort were given by Weierstrass in 1872 who gave an
example involving an infinite series in which each term had all derivatives everywere
and yet the uniform limit had no derivative anywhere. Using the example of Theorem
2.2.41, give an example of an infinite series of functions, each term being a polynomial
defined on ,
f for which it makes absolutely no sense to write
f′ fails to exist at any point. In other words, you cannot
differentiate an infinite series term by term. The derivative of a sum is not the sum of
the derivatives when dealing with an infinite “sum”. Also show that if you have any
differentiable function g and ε > 0, there exists a nowhere differentiable function h such that
< ε. This is in stark contrast with what will be presented in complex analysis in which,
thanks to the Cauchy integral formula, uniform convergence of differentiable functions does
lead to a differentiable function. Hint: Use Weierstrass approximation theorem and
telescoping series to get the example of a series which can’t be differentiated term by
- Consider ℝ ∖
. Show this is not connected.
- Show S ≡
∪ is connected but not arcwise connected.
- Let A be an m × n matrix. Then A∗, called the adjoint matrix, is obtained from A by taking the
transpose and then the conjugate. For example,
ij = Aji. Show that =
. The inner product is
described in the chapter. Recall
- Let X be a subspace of Fm having dimension d and let y ∈ Fm. Show that x ∈ X is closest to y in
the Euclidean norm out of all vectors in
X if and only if = 0 for all
u ∈ X. Next show
there exists such a closest point and it equals ∑
j=1d an orthonormal basis
- Let A : Fn → Fm be an m × n matrix. (Note how it is being considered as a linear transformation.)
≡ is a subspace of
Fm. If y ∈ Fm is given, show that there exists x such
that y −Ax is as small as possible (Ax is the point of Im closest to
y) and it is a solution to the
least squares equation
Hint: You might want to use Problem 24.
- Show that the usual norm in Fn given by
satisfies the following identities, the first of them being the parallelogram identity and the second
being the polarization identity. Show that these identities hold in any inner product space, not just Fn. By definition, an inner
product space is just a vector space which has an inner product.
- Let K be a nonempty closed and convex set in an inner product space which is
complete. For example,
Fn or any other finite dimensional inner product space. Let y
Let be a minimizing sequence. That is
Explain why such a minimizing sequence exists. Next explain the following using the parallelogram
identity in the above problem as follows.
Hence Next explain why the right hand side converges to 0 as m,n →∞. Thus is a Cauchy sequence
and converges to some
x ∈ X. Explain why x ∈ K and =
λ. Thus there exists a closest point
in K to y. Next show that there is only one closest point. Hint: To do this, suppose there are two
x1,x2 and consider using the parallelogram law to show that this average works better than
either of the two points which is a contradiction unless they are really the same point. This theorem
is of enormous significance.
- Let K be a closed convex nonempty set in a complete inner product space (Hilbert space) and
y ∈ H. Denote the closest point to y by Px. Show that Px is characterized as being the solution
to the following variational inequality
for all z ∈ K. Hint: Let x ∈ K. Then, due to convexity, a generic thing in K is of the form
x + t
,t ∈ for every
z ∈ K. Then
If x = Py, then the minimum value of this on the left occurs when t = 0. Function defined on
has its minimum at
t = 0. What does it say about the derivative of this function at t = 0? Next
consider the case that for some x the inequality Re
≤ 0. Explain why this shows
x = Py.
- Using Problem 29 and Problem 28 show the projection map, P onto a closed convex subset is
Lipschitz continuous with Lipschitz constant 1. That is
- Let A : ℝn → ℝn be continuous and let f ∈ ℝn. Also let denote the standard inner product in
ℝn. Letting K be a closed and bounded and convex set, show that there exists x ∈ K such that for all
y ∈ K,
Hint: Show that this is the same as saying P=
x for some x ∈ K where here
P is the projection map discussed above. Now use the Brouwer fixed point theorem.
This little observation is called Browder’s lemma. It is a fundamental result in nonlinear
- ↑In the above problem, suppose that you have a coercivity result which is
Show that if you have this, then you don’t need to assume the convex closed set is bounded. In case
K = ℝn, and this coercivity holds, show that A maps onto ℝn.