For what it's worth, I'll contribute what I think the Hebrews passage is saying.
HEB 11:1 Now faith is the substance of things hoped for, the evidence of things not seen. (KJV)
In Heb. 1:1, one's faith IS the proof or evidence that there is a God and a heavenly realm. In other words, one in possession of faith is in possession of something beyond this world.
The Free Online Dictionary defines faith as
1. Confident belief in the truth, value, or trustworthiness of a person, idea, or thing.
2.
Belief that does not rest on logical proof or material evidence.
In the general sense, when we speak of believing something, we mean that we are accepting as true some piece of information that we do not have the kind of evidence or proof for that we would normally require to incorporate it into our picture of reality. When we have faith, it means we believe this information to be true
We think of believing (and, therefore, having faith, since they come from the same word in Greek) as accepting some piece of information as true. Sometimes it's because new information fits with what we have already accepted. Other times, we may accept the information based on our trust of the source (expert, authority figure, someone with established reputation as trustworthy, etc.) Whatever the case, it's an intellectual process—it takes place in the brain. This can also be accompanied by emotion—believing something may produce an emotional reaction, at least initially. In fact, the emotions will create a stronger imprint. So, faith, in this sense, is purely mental, but it can produce an emotional reaction and it can affect our behavior as we adjust to conform (consciously or unconsciously) to the new reality. This kind of mental faith ebbs and flows, and can be modified with the acceptance of new information. There's no guaranteed permanence to it. You have to reinforce it and protect it.
I see something else in this passage. If the passage was referring to this kind of faith, what kind of evidence is that? It's no different than any other kind of belief. All it shows is that the person is a normal, thinking human being. No, I think the faith referred to here is different. I think it's meant to refer to the faith that God gives through the Holy Spirit—a faith that is spiritual, not mental, or even emotional. That faith is a connection with God, and it is the ability to believe God for even incredible things, but it is not the result of human effort, brainwashing, positive thinking. To have that kind of faith—what I call the "faith of God" (re: preferred translation of Mark 11:22)—is the evidence this verse refers to. It's given by God through Christ, and is the evidence of what has been promised.
You may or may not agree that the theology in this statement is true, but I think that is what the writer intended.