Let H1,H2 be Hilbert spaces and T be a closed linear operator defined on a dense subspace D(T) in H1 and taking values in H2. In this article we prove the following results: (i) Range of T is closed if and only if 0 is not an accumulation point of the spectrum σ(T*T) of T*T, In addition, if H1 = H2 and T is self-adjoint, then (ii) inf {{double pipe}Tx{double pipe}:x∈D(T)∩N(T)⊥{double pipe}x{double pipe} = 1} = inf{|λ|: 0 ≠ λ ∈ σ(T)}, (iii) Every isolated spectral value of T is an eigenvalue of T (iv) Range of T is closed if and only if 0 is not an accumulation point of the spectrum σ(T) of T (v) σ(T) bounded implies T is bounded. We prove all the above results without using the spectral theorem. Also, we give examples to illustrate all the above results.