- 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 spanis closed. That is, the span of any finite set of vectors is always a closed subspace. Hint: Suppose you let V = spanand let v
^{n}→ 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≡is NEVER compact even though it is closed and bounded. Hint: Suppose you have
_{i=1}^{n}where≥. Let yspan, a closed subspace. Such a y exists because X is not finite dimensional. Explain why dist> 0. This depends on spanbeing closed. Let z ∈ spansuch that≤ 2dist. Let x_{n+1}≡. Then consider the following: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 ℝ
^{2}. Hint: Maybe make it easy on yourself and consider diagonal matrices. - Now let V = Cand 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, B⊆ U. This says, in topological terms, thatis regular. Is it always the case in a metric space that B=≡ D? Prove or disprove. Hint: In fact, the answer to the last question is no.
- Let be a complete metric space. Letbe a sequence of dense open sets. This means that B∩ U
_{n}≠∅ for every x ∈ X, and r > 0. You know that ∩_{n}U_{n}is not necessarily open. Show that it is nevertheless, dense. Hint: Let D = ∩_{n}U_{n}. You need to show that B∩ D≠∅. There is a point p_{1}∈ U_{1}∩ B. Then there exists r_{1}< 1∕2 such that B⊆ U_{1}∩B. From the above problem, you can adjust r_{1}such that B⊆ U_{1}∩ B. Next there exists p_{2}∈ B∩ U_{2}. Let r_{2}< 1∕2^{2}be such that B⊆ B∩ U_{2}∩ U_{1}. Continue this way. You get a nested sequence of closed setssuch that the diameter of B_{k}is no more than 1/2^{k−1}, the k^{th}being contained in B∩∩_{i=1}^{k−1}U_{i}. Explain why there is a unique point in the intersection of these closed sets which is in B∩∩_{k=1}^{∞}U_{k}. Then explain why this shows that D is dense. - The countable intersection of open sets is called a G
_{δ}set. Show that the rational numbers ℚ is NOT a G_{δ}set in ℝ. In fact, show that no countable dense set can be a G_{δ}set. Show that ℕ is a G_{δ}set. It is not dense. - You have a function f : →. Define
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 G_{δ}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 ℝ
^{n}∖A is connected. You might consider the case n > 1 and the case n = 1 separately. - If A is connected in ℝ
^{n}, then ℝ^{n}∖ A is also connected. - If A is connected in ℝ
^{n}, then either A is open or A is closed. - ℝ
^{n}∖ Bis connected. Two cases to consider: n = 1 and n > 1.

- If A is compact, then ℝ
- If A is a connected set in ℝ
^{n}, 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 deletingand then taking the two closed intervals which result and deleting the middle open third of each of these and continuing this way. Let J
_{k}denote the union of the 2^{k}closed intervals which result at the k^{th}step of the construction. The Cantor set is J ≡∩_{k=1}^{∞}J_{k}. 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 ontoeven 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 x. - You have a complete metric space and a mapping T : X → X which satisfies
Consider the sequence, x,Tx,T

^{2}x,. Show that this converges to a point z ∈ X such that Tz = z. Next suppose you only know< R and that on B,d≤ rdwhere r < 1 as above. Show that then z ∈ Band that in fact each T^{k}x ∈ B. Show also there is no more than one such fixed point z on B. - 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 ∑
_{|α| ≤m}a_{α}x^{α}where a_{α}∈ 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
that
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 showis a continuous function. - Find an example of a sequence of continuous functions defined on ℝ
^{n}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 ℝ^{n}∖, that is, the set of vectors in ℝ^{n}excluding 0. - An open subset U of ℝ
^{n}is arcwise connected if and only if U is connected. Consider the usual Cartesian coordinates relative to axes x_{1},,x_{n}. 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 ℝ^{n}is connected if and only if every two points can be joined by a square curve. - Let x → hbe a bounded continuous function. Show the function f= ∑
_{n=1}^{∞}is continuous. - Let S be a any countable subset of ℝ
^{n}. Show there exists a function, f defined on ℝ^{n}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 test. - By Theorem 3.12.7 there exists a sequence of polynomials converging uniformly to
f=on R ≡∏
_{k=1}^{n}. Show there exists a sequence of polynomials,converging uniformly to f on R which has the additional property that for all n,p_{n}= 0 . - If f is any continuous function defined on K a sequentially compact subset of ℝ
^{n}, show there exists a series of the form ∑_{k=1}^{∞}p_{k}, where each p_{k}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≡ distwhere S is a nonempty subset of ℝ
^{n}. Show f is uniformly continuous. HereOf 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 fis also sequentially compact.
- If f is uniformly continuous, does it follow that is also uniformly continuous? Ifis uniformly continuous does it follow that f is uniformly continuous? Answer the same questions with “uniformly continuous” replaced with “continuous”. Explain why.

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