Supplementary Linear Algebra textbook

Linear Algebra Done Right by Sheldon Axler is a very clear book that takes a proof based approach to linear algebra and works natively with vector spaces with a minimum of algebraic matrix manipulation. I highly recommend it.

The book Linear Algebra by Friedberg, Insel and Spence is a very comprehensive book with lots of examples and exercises. You can find some examples of infinite dimensional vector spaces in the book. It even includes optional topics like dual vector spaces, which serves as a nice initiation to linear functional-analysis.

A completely different option would be to use the materials for the Massive Open Online Course titled "Linear Algebra: Foundations to Frontiers" on edX. It has online exercises, video, and simple programming exercises that links abstraction in mathematics to abstraction in programming. It is meant to be more appealing to students who are more computational science/computer science oriented.

This course does not require calculus and does include proofs of essentially all results.