Unbelief is not necessarily failure to acknowledge the truth. You can acknowledge or assent to something being true and still be unbelieving. UNBELIEVING IS MISTRUST. It is failure to trust a person. Abraham believed God, yet there were times when he did not trust Him. Twice he let Sarah go into a harem because he did not trust God to take care of him. Unbelief is actually the sin from which all other sins come! This is what Satan used to tempt Eve. "Hath God said..." Satan did not mean, "Did God really make that statement?" He meant, "You don't have to believe what He said implicitly." "You shall not surely die." The stress and worry we have is nothing more than unbelief! It is a failure to trust God! The lack of assurance we have is unbelief. It is failure to trust Christ. From this sin (unbelief) comes all other sins!
"How can a believer have unbelief?" someone asks. Because of the flesh. An unbeliever has nothing but unbelief. A believer has faith; but he still has unbelief and has need to cry daily, "Lord, I believe, help Thou mine unbelief!"