- It was shown that in a finite dimensional normed linear space that the compact
sets are exactly those which are closed and bounded. Explain why every finite
dimensional normed linear space is complete.
- In any normed linear space, finite dimensional or not, show that span
is closed. That is, the span of any finite set of vectors is always a closed subspace.
Hint: Suppose you let V = span and let
vn → v be convergent
sequence of vectors in V . What does this say about the coordinate maps?
Remember these are linear maps into F and so they are continuous.
- It was shown that in a finite dimensional normed linear space that the compact sets are
exactly those which are closed and bounded. What if you have an infinite dimensional
normed linear space X? Show that the unit ball D
compact even though it is closed and bounded. Hint: Suppose you have
, a closed subspace. Such a y exists
because X is not finite dimensional. Explain why dist
> 0. This
depends on span being closed. Let
z ∈ span such that
. Let xn+1 ≡. Then consider the
What of ? Where is it? Isn’t it in span
? Explain why this
yields a sequence of points of
X which are spaced at least 1/2 apart even though they
are all in the closed unit ball.
- Find an example of two 2 × 2 matrices A,B such that
. This refers to
the operator norm taken with respect to the usual norm on
ℝ2. Hint: Maybe make it
easy on yourself and consider diagonal matrices.
- Now let V = C and let
T : V → V be given by
Show that T is continuous and linear. Here the norm is
Can you find where this is the operator norm defined by analogy to what was given
in the chapter?
- Show that in any metric space
, if U is an open set and if x ∈ U, then
there exists r > 0 such that the closure of B
⊆ U. This says, in
topological terms, that is regular. Is it always the case in a metric space that
≡ D? Prove or disprove.
Hint: In fact, the answer to
the last question is no.
- Let be a complete metric space. Let
be a sequence of dense open sets. This
∩ Un≠∅ for every x ∈ X, and r > 0. You know that ∩nUn is not
necessarily open. Show that it is nevertheless, dense. Hint: Let D = ∩nUn. You need
to show that B
∩ D≠∅. There is a point p1 ∈ U1 ∩ B
. Then there exists
r1 < 1∕2 such that B
⊆ U1 ∩B
. From the above problem, you can adjust
r1 such that B
⊆ U1 ∩ B
. Next there exists p2 ∈ B
∩ U2. Let
r2 < 1∕22 be such that B
∩ U2 ∩ U1. Continue this way. You get a
nested sequence of closed sets such that the diameter of
Bk is no more than
1/2k−1, the kth being contained in B
∩∩i=1k−1Ui. Explain why there is a
unique point in the intersection of these closed sets which is in B
Then explain why this shows that D is dense.
- The countable intersection of open sets is called a Gδ set. Show that the rational
numbers ℚ is NOT a Gδ set in ℝ. In fact, show that no countable dense set can be a
Gδ set. Show that ℕ is a Gδ set. It is not dense.
- You have a function f :
Then explain why
exists. Explain why a function is continuous at x if and only if ωf = 0. Next show
that the set of all
x where ωf = 0 is a
Gδ set. Hint: ωf = 0 if and only if
x is in
something like this: ∩n=1∞∪k=1∞
. Explain this. Then explain why
∪k=1∞ is an open set.
- Prove or disprove.
- If A is compact, then ℝn∖A is connected. You might consider the case n > 1
and the case n = 1 separately.
- If A is connected in ℝn, then ℝn ∖ A is also connected.
- If A is connected in ℝn, then either A is open or A is closed.
- ℝn ∖ B is connected. Two cases to consider:
n = 1 and n > 1.
- If A is a connected set in ℝn, and A is not a single point, show that every point of A is
a limit point of A.
- Consider the Cantor set. This is obtained by starting with deleting
then taking the two closed intervals which result and deleting the middle open third of
each of these and continuing this way. Let
Jk denote the union of the 2k closed
intervals which result at the kth step of the construction. The Cantor set is
J ≡∩k=1∞Jk. Explain why J is a nonempty compact subset of ℝ. Show
that every point of J is a limit point of J. Also show there exists a mapping
from J onto even though the sum of the lengths of the deleted open
intervals is 1. Show that the Cantor set has empty interior. If
x ∈ J, consider
the connected component of x. Show that this connected component is just
- You have a complete metric space and a mapping
T : X → X which
Consider the sequence, x,Tx,T2x,
. Show that this converges to a point
z ∈ X such that Tz = z. Next suppose you only know
< R and that on
≤ rd where
r < 1 as above. Show that then z ∈ B and
that in fact each
Tkx ∈ B. Show also there is no more than one such fixed point
- In Theorem 3.11.5 it is assumed f has values in F. Show there is no change if f has
values in V, a normed vector space provided you redefine the definition of a polynomial
to be something of the form ∑
≤maαxα where aα ∈ V .
- How would you generalize the conclusion of Corollary 3.12.8 to include the situation
where f has values in a finite dimensional normed vector space?
- If f and g are real valued functions which are continuous on some set, D, show
are also continuous. Generalize this to any finite collection of continuous functions.
Hint: Note max =
. Now recall the triangle inequality which can be
used to show is a continuous function.
- Find an example of a sequence of continuous functions defined on ℝn such that each
function is nonnegative and each function has a maximum value equal to 1 but the
sequence of functions converges to 0 pointwise on ℝn ∖
, that is, the set of vectors in
ℝn excluding 0.
- An open subset U of ℝn is arcwise connected if and only if U is connected. Consider the
usual Cartesian coordinates relative to axes x1,
,xn. A square curve is one consisting
of a succession of straight line segments each of which is parallel to some coordinate
axis. Show an open subset U of ℝn is connected if and only if every two points can be
joined by a square curve.
- Let x → h be a bounded continuous function. Show the function
- Let S be a any countable subset of ℝn. Show there exists a function, f defined on ℝn
which is discontinuous at every point of S but continuous everywhere else.
Hint: This is real easy if you do the right thing. It involves the Weierstrass M
- By Theorem 3.12.7 there exists a sequence of polynomials converging uniformly to
. Show there exists a sequence of polynomials,
converging uniformly to
f on R which has the additional property that for all
n,pn = 0
- If f is any continuous function defined on K a sequentially compact subset of ℝn, show
there exists a series of the form ∑
k=1∞pk, where each pk is a polynomial, which
converges uniformly to f on .
Hint: You should use the Weierstrass approximation
theorem to obtain a sequence of polynomials. Then arrange it so the limit of this
sequence is an infinite sum.
- Consider f
≡ dist where
S is a nonempty subset of ℝn. Show f is uniformly
Of course is some norm.
- Let K be a sequentially compact set in a normed vector space V and let f : V → W be
continuous where W is also a normed vector space. Show f is also sequentially
- If f is uniformly continuous, does it follow that is also uniformly continuous? If
uniformly continuous does it follow that
f is uniformly continuous? Answer the same
questions with “uniformly continuous” replaced with “continuous”. Explain