Banach Fixed Point Theorem

Related BrainMass Solutions

• Hahn Banach Theorem Application

  36860 Hahn Banach Theorem Application Suppose that is a Banach space over K.

• Banach Space and Hamel Dimension

  Theorem 1. Suppose that E is a Banach space. Prove that the Hamel dimension of E is not In order to prove the above statement, it suffices to prove the following. Theorem 2.

• Normed Linear Space : Hahn-Banach Theorem

  If M E, then apply the Hahn-Banach Theorem. A Normed Linear Space and the Hahn-Banach Theorem are investigated. The solution is detailed and well presented.

• Dual Space and Isometrically Isomorphic Spaces

  It defines a natural map and uses the Hahn Banach theorem and determines if c* and c are isometrically isomorphic.

• Two-part question on a finite dimensional normed linear space

  Theorem: (i) since closed and bounded subset of a banach space is compact, the closed unit ball is compact.

• Mapping, Contraction and Fixed-Point Theorem : Let T(x) = x^2 Show that T is a contraction on (0, 1/3] , but that T has no fixed point on this interval. Does this conflict Theorem 6.4? Explain.

  Then T must have a unique fixed point. This is what the Theorem 6.4 tells all about the fixed point. Now, let's verify the validity of the theorem. Suppose T has a fixed point, Then T(x) = x.

• Prove that " ".

  35553 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 .]

• Intermediate value theorem

  420887 Fixed Point Theorem (using the Mean Value Theorem) Prove that if f is a differentiable function with f´(x) > 1 for all x, then there is at most one c such that f(c) = c.

• Proof of Fixed Point Theorem using Stokes Theorem and Analysis

  18931 Proof of Fixed Point Theorem using Stokes Theorem and Analysis Prove that if D is the closed disc |x|  1 in R2, then any map f 2 C2[D ! D] has a fixed point: f(x) = x. The proof is by contradiction, and uses Stokes theorem.