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.

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*.

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.

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.