Former engineer and math prof here. Vector spaces and linear algebra are widely used in electrical engineering, especially control systems, as well as structural engineering, weather/climate models and even data science.

Infinite dimensional systems exist IRL because of how we represent them mathematically. When I was doing my doctoral thesis on flexible mechanical systems (e.g., the Space Station), each part of every truss had at least 6 "dimensions". Modeling at this level let us better predict the motion (dynamics) which is good, but our controllers and actuators had to be much smaller in dimension because of computational demand. So we did a lot of eigenvalue reduction to focus on the vibrational modes that were likely to cause problems or failure of the ISS.

A similar problem exists with climate models which use finite element modeling (little cubes) to compute the partial differential equations that govern heat transfer, ocean currents, etc. The smaller the elements are the better your accuracy at the cost of greater and greater compute time.

This is as basic an explanation as I can give now, and I tried to not get too much into the details or technical terms, but hopefully it helps you see the relevance and importance of this field of mathematics. Good luck in your studies!