The aim of this paper is to introduce and give preliminary investigation of ∂-partial-locally COMPACT SPACEs. Locally COMPACTness and ∂-partial-locally COMPACTness are independent of each other. Every locally COMPACT Hausdorff SPACE is ∂-partial-locally COMPACT. But the converse is not true even though it be Hausdorff. ∂-partial-locally COMPACTness is a topological property. ∂-partial-locally COMPACTness is not preserved by the product topology.

Let (M,g) be a COMPACT immersed hypersurface of (Rn+1,<,>), l1 the first nonzero eigenvalue, a the mean curvature, P the support function, A the shape operator, vol(M) the volume of M, and S the scalar curvature of M. In this paper, we established some eigenvalue inequalities and proved the above.1/n òM \\A\\2 P2DV ³ </span></p>M  a2 P2 dV. 2 òM a2 P2 dV ³ 1/n (n-1) òM sp2 dV.If the scalar curvature S and the first nonzero eigenvalue l1 staisfy S= l1 (N-1), than òM [a2 - l1/n] p2 dv³ 0, 4) Suppose that the Ricci curvature of M is bounded below by a positive constant k. ThusòM a2 P2 dV ³ k/n(n-1) òM \\ gradf\\ dv+vol(M). 5) Suppose that the Ricci curvature is bounded and the scalar curvature satisfy S= l1(n− 1) and L=k- 2S>0 is a constant. Thusvol (M) ³ kl1 /L òM\\ Y \\ a pdv – 2S/L òM a2 P2 Dv.

In this paper, we give the characterization of some classes of COMPACT operators given by matrices on the normed sequence SPACE rqp (Bm), which is a special case of the paranormed Riesz Bm-difference sequence SPACE rq (p, Bm). For this purpose, we apply the Hausdorff measure of nonCOMPACTness and use some results.

For suitable Banach SPACEs X and Y with Schauder decompositions and a suitable closed subSPACE M of some COMPACT operator SPACE from X to Y, it is shown that the strong Banach-Saks-ness of all evaluation operators on M is a sufficient condition for the weak Banach-Saks property of M, where for each xÎX and y*ÎY*, the evaluation operators on M are defined by fx (T)= Tx and yy* (T)= T*y*.

We show that typical elements of the set of continuous functions from a COMPACT differentiable manifold M to Rn are nowhere differentiable. Then, we study the box dimensions of typical elements in the set of images of M in Rn.