MTH641 Assignment 1 Solution Fall 2026

This assignment covers fundamental concepts in functional analysis, including metric spaces, continuous mappings, sequence spaces, and completeness. Students are required to provide formal proofs and mathematical arguments for various properties related to these topics. The tasks involve analyzing specific metric definitions, identity mappings, density in sequence spaces, and the convergence of subsets within complete metric spaces.

1. 1

   Question 1: Metric Spaces and Open Sets

   5 marks

   Part (a) requires proving that a specific function d is a metric on R. Part (b) asks to determine two specific open balls, Bd(0, 1/2) and Bd(0, 1). Part (c) asks whether the set A = (0, 1) is open in (R, d) and requires justification.

2. 2

   Question 2: Continuous Mappings

   5 marks

   Part (a) asks for a proof that the inverse image of an open set under a continuous function is open. Part (b) requires determining the continuity of two identity mappings between R equipped with the discrete metric and R equipped with the standard metric, providing justification for both.

3. 3

   Question 3: Sequence Spaces and Separability

   5 marks

   Part (a) asks to show that the set D of sequences with rational coordinates and finitely many nonzero terms is countable. Part (b) requires proving that D is dense in the space l^2 to conclude it is separable. Part (c) asks to show that the sequence (en) is bounded but not a Cauchy sequence in l^2.

4. 4

   Question 4: Convergence and Completeness

   5 marks

   Part (a) and (b) require proving the relationship between closed subsets and completeness in a complete metric space. Part (c) asks for a Cauchy sequence in the subset Y = (0, 1) that does not converge within Y and an explanation of why Y is not complete.

What to submit

One Microsoft Word document (.doc or .docx).

Topics covered

Metric spaces Open sets Continuous mappings Sequence spaces Separability Completeness Cauchy sequences