The gym was built 15 feet below the main house. It does not have radiant heat and attached to the main house. The gym has a service entrance on gym floor level, exterior entrance and interior entrance. The gym is cooled and heated with multiple units, Efficiency Mechanical Inc. would have the details. Thank you.

I've seen this in Utah many times. Often under a garage or in a sub basement.

You could heat this space with radiant heat, but it would require heat loss calculations to make sure enough heat was planned for (as would any heating system).

Sport courts can be added to homes in various ways. Here in Utah along the wasatch front many homes are built on hillsides and this often provides opportunities for high basements that can be ideal for this use. Whether the space is connected or closed off is a personal choice. They often are closed just to help keep down the noise transfer. But an overlook can still be provided that is either part of the court or overlooks through observation windows. We have designed these types of rooms under many garages and usually those rooms tend to be more closed off. Another option is to build down as far as is reasonable, say below the basement level as another comment stated and then extending up to the desired height.