Faith is often thought of as "believing in something we cannot see," or "believing in God." That's almost become the textbook definition, even to an unbeliever. The discussion gets interesting though, when we talk about "saving faith" being more than just believing something exist. We've heard it a thousand times, that even the devil believes there's a God. So the manuscripts lead us to the word "believing" meaning trusting, relying in Him.
But we were talking about faith. "Faith is the assurance of things hoped for, and conviction of things not seen." (Heb 11:1) Things that we hope for; the empty tomb, eternal life, transformation to glorification, having peace in a fallen world, and more, are all a done deal. We are certain. We are sure. There's no doubt. End of story.
So is there such thing as "saving faith?" I think "faith" covers it all. And it was free. A gift from God. (Rom 12:3) Proof of grace to all who have it.