Study-guide for final exam

• Purpose of the Study Guide:
  • Summarizes problems from various homework sets covered during the term.
• Final Exam Information:
  • One problem will be selected from each homework set.
  • The selected problems may be similar but not identical to those listed in the guide.
• Preparation Recommendations:
  • Revisit each problem from the guide.
  • Focus on understanding the underlying concepts.
  • Be prepared to solve variations of the problems.
• Study Goals:
  • Use the guide to review:
  • Definitions
  • Important results
  • Proofs
  • Problem-solving techniques
• Topics Covered:
  • Problems are grouped by homework sets.
  • Topics include:
  • Vector spaces
  • Metric spaces
  • Normed spaces
  • Hilbert spaces

Problem 1: Prove that the set of all continuous functions on [0, 1] is a vector space.

Problem 2: Prove that the vector space of all continuous functions on [0,1] is infinite-dimensional.

Problem 3: Find a basis for the vector space
\[
V = \left\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} : a,b,c,d \in \mathbb{C}\right\}
\]
and compute its dimension.

Problem 4: Consider \( g(t)=e^{-t^2} \). For \( x \in \mathbb{R} \), define
\((T_x g)(t) = g(t – x)\). Let \(\Lambda\) be a finite subset of \(\mathbb{R}\). Prove that
\(\{T_\lambda g : \lambda \in \Lambda\}\) is linearly independent in the space of all continuous functions on \(\mathbb{R}\).

Problem 1: Prove that \( d(x,y)=|x-y| \) defines a metric on \(\mathbb{R}\).

Problem 2: Consider \(X=\mathbb{R}^2\) with the Euclidean metric. Describe the metric inherited by
\(Y = \{(x,0): x \in \mathbb{R}\}\) and what it looks like.

Problem 3: Compute the discrete metric for \(X=\{1,2,3,4\}\).

Problem 4: For \(x=(1,2)\) and \(y=(4,6)\) in \(\mathbb{R}^2\), calculate \(d_1(x,y)\), \(d_2(x,y)\), and \(d_{\infty}(x,y)\).

Problem 5: Consider \(\ell^1\), the space of absolutely summable sequences. Determine if various given sequences belong to \(\ell^1\) and find sufficient conditions for membership.

Problem 6: For \(x,y \in \ell^1\), show \(d_1(x,y)=\sum_{k=1}^{\infty}|x_k – y_k|\) is a metric.

Problem 7: Consider function spaces on [0,1] with uniform and \(L^1\) metrics. Prove metrics are well-defined and compare distances and convergence in these metrics.

Problem 1: Determine convergence of sequences in \((\mathbb{R},|\cdot|)\). Discuss sequences like \(x_n=1/n\) and \(x_n=(-1)^n\).

Problem 2: Consider \(f_n(t)=t^n\) in \(C([0,1])\) under the \(L^1\)-metric and the uniform metric. Investigate convergence in each metric and compare.

Problem 3: Repeat a similar investigation as above but possibly with a modified sequence \(f_n(t)=n t\) or other variants, analyzing their convergence in different metrics.

Problem 4: Show that \(C([0,1])\) with the \(L^1\)-metric is not complete by constructing a Cauchy sequence that converges to a non-continuous limit.

Problem 5: Prove that \(\mathbb{R}\) is complete and that \(\ell^1\) is complete, showing every Cauchy sequence has a limit in those spaces.

Problem 1: Investigate open sets in the discrete metric. Show every subset is open and describe convergence of sequences in the discrete metric.

Problem 2: Characterize open balls in \(\mathbb{R}^2\) with the \(d_1\), \(d_2\), and \(d_\infty\) metrics (shapes: diamond, circle, square).

Problem 3: Prove general properties of interiors of sets: \(A^\circ\), closure, and the relationship with open/closed sets and intersections.

Problem 1: Define the following terms in your own words:

(a) Closed set in a metric space,

(b) Accumulation point of a set,

(c) Boundary point of a set.

Problem 2: Take \( X = \mathbb{R} \) and find \( E^\circ \), \( \partial E \), and \( \overline{E} \) for each of the following sets:

(a) \( E = \left\{ 1, \frac{1}{2}, \frac{1}{3}, \dots \right\} \)

(b) \( E = [0, 1) \)

(c) \( E = \mathbb{Z} \), the set of integers.

(d) \( E = \mathbb{Q} \), the set of rationals.

(e) \( E = \mathbb{R} \setminus \mathbb{Q} \), the set of irrationals.

Problem 3: Let \( E \) be a subset of a metric space \( X \). Show that if \( E \) is not dense in \( X \), then its complement must contain an open ball.

Problem 4: Let \( E \) be a subset of a metric space \( X \). Prove that \( E \) has empty interior if and only if its complement is dense in \( X \).

Problem 5: Given a subset \( E \) of a metric space \( X \), prove the following statements:

(a) \( E \) is closed if and only if \( E = \overline{E} \).

(b) \( \overline{E} \) is the union of \( E \) and all of the accumulation points of \( E \).

(c) \( \overline{E} = E \cup \partial E \).

Problem 6: Prove that the set of all finite sequences is not a dense subset of \( \ell^{\infty} \).

Problem 1: Prove that arbitrary intersections and finite unions of compact sets are compact.

Problem 2: Show that if \(K\) is compact in \(\mathbb{R}\), then \(K \times \{0\}\) is compact in \(\mathbb{R}^2\) with the standard metric.

Problem 1: Show \(f:X \to Y\) is continuous if and only if the preimage of every closed set is closed.

Problem 2: If \(E\) is dense in \(X\) and \(f,g:X \to Y\) are continuous with \(f=g\) on \(E\), then \(f=g\) on \(X\).

Problem 3: Prove that a uniformly continuous function sends Cauchy sequences in \(X\) to Cauchy sequences in \(Y\).

Problem: Openness of Sets in a Metric Space

Let \( E \) and \( F \) be disjoint closed subsets of a metric space \( X \). For each point \( x \in X \), define the distance functions:

\[
f(x) = \text{dist}(x, E) \quad \text{and} \quad g(x) = \text{dist}(x, F).
\]

Consider the sets:

\[
U = \{ x \in X : f(x) < g(x) \} \quad \text{and} \quad V = \{ x \in X : g(x) < f(x) \}. \]

Prove that \( U, V \) are disjoint open sets satisfying \( E \subseteq U \) and \( F \subseteq V \).

Problem 1: In \((\mathbb{R}^n,\|\cdot\|_p)\), prove componentwise convergence from norm convergence.

Problem 2: In \((\ell^p,\|\cdot\|_{\ell^p})\), prove componentwise convergence from norm convergence.

Problem 3: Show that the norm function \(x \mapsto \|x\|\) is continuous.

Problem 4: If a subspace \(V\) of a normed vector space \(X\) is open (in the induced metric), prove \(V=X\).

Problem 1: Give a Cauchy sequence in \(\ell^1\) and prove it converges in \(\ell^1\).

Problem 2: Prove that \(c_{00}\) (sequences with finite support) is dense in \(\ell^p\).

Problem 3: Show that the closure of \(c_{00}\) in the \(\ell^\infty\)-norm is \(c_0\).

Problem 1: Prove the 5-norm and the 3/4-norm on \(\mathbb{R}^2\) are equivalent by finding constants \(m,M\).

Problem 2: Show two norms are equivalent if and only if they induce the same topology.

Problem 3: In \(c_{00}\), show the norms associated with \(\ell^1\) and \(\ell^2\) are not equivalent.

Problem 1: In an inner product space, prove uniqueness property (if \(\langle x,z\rangle=\langle y,z\rangle\) for all \(z\), then \(x=y\)) and the characterization of the norm via the sup definition.

Problem 2: Show \(H \times K\) is a Hilbert space if \(H,K\) are Hilbert spaces.

Problem 3: Given a positive definite matrix \(A\), define \(\langle x,y\rangle_A=(Ax)\cdot y\). Prove it is an inner product.