Analysis - Limit of the Average of the first N terms of a Sequence

Proving in Real Analysis Course. Please see the attached.

Banach Space Isomorphism Prove that “ ”, that is, prove that there is a Banach space isomorphism [Hint: for each in , define for each in Show that and .]

Functional Analysis Banach space ...continues

Bounded Linear Operator; Bounded Invertible

Please solve the attached problems on bounded linear operators and bounded invertible equations.

Hahn Banach Theorem Application

Please assist me with the attached Hahn Banach Theorem problem. Thanks very much.

E is Nomed Linear Space ... Hahn-Banach Theorem.

Suppose that E is a normed linear space. Prove that if E* is separable, then E is separable. **See attachment for complete problem. Thanks!

Normed Linear Spaces Suppose that ε and F are normed linear spaces and ε ≠ 0. Prove that ℓ ( ε, F ) is complete, then is complete.

Functional Analysis Normed Linear Space ...continues

Disease Data - Rectangular Coordinate system

1985 1990 1995 2000 Heart Disease 778375 727206 737,563 710760 Cancer 459121 510426 538,455 1220100 AIDS 1700 25370 43115 14999 I do not understand how to plot data. I need to do this f ...continues

Weak Convergence

Suppose that E is a normed linear space, and C is a subset. Prove that C is weakly bounded if and only if C is norm bounded. Conclude that weakly convergent sequences in E are bounded.

Continuity of a linear functional on a topology.

Suppose that E is a vector space, L is a family of linear functionals on E , and g is a linear functional on E. Let t be the topology on E induced by L. Prove that g is t-continuous on E if and only if g is a linear combination of L.

Suppose that ε is a normed linear space. Let j: ε → ε** be the canonical imbedding and let x** be a linear functional on ε*. Then x** is weak* continuous if and only if x** Є j(ε).

Topology Suppose that ε is a normed linear space. Let j: ε → ^ ...continues