# Real Analysis

1. Consider the set of real numbers ℝ. Say that xy (mod 2π) if xy is an integer multiple of 2π. Verify that this is an equivalence relation.

2. Put a relation on C[0, 1] by fg if f(k/10) = g(k/10) for 0 ≤ k ≤ 10.

• (a) Verify that this is an equivalence relation.
• (b) Describe the equivalence classes.
• (c) Show that [f] + [g] = [f + g] is a well-defined operation.
• (d) Show that t[f] = [tf] is well defined for all t ∈ ℝ and fC[0, 1].
• (e) Show that these operations make C[0, 1]/∼ into a vector space of dimension 11.

3. Say that two real vector spaces V and W are isomorphic if there is an invertible linear map T of V onto W.

• (a) Prove that this is an equivalence relation on the collection of all vector spaces.
• (b) When are two finite-dimensional vector spaces isomorphic?

4. Define |x| = max{x, −x}.

• (a) Prove that |xy| = |x| |y| and |x1| = |x|1.
• (b) Prove the triangle inequality |x + y| ≤ |x| + |y|.

5. Prove that if x < y, then there is a rational number r with a finite decimal expansion such that x < r < y.

6. Prove that if x < y, then there is an irrational number z such that x < z < y.