Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem
                Multivariable Calculus Review
  Multivariable Calculus Review
  Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem
       Multi-Variable Calculus
       Point-Set Topology
       Compactness
       The Weierstrass Extreme Value Theorem
       Operator and Matrix Norms
       Mean Value Theorem
  Multivariable Calculus Review
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     ◮                       n
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        ν(x) ≥ 0 ∀ x ∈ R with equality iff x = 0.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     ◮ ν(αx) = |α|ν(x) ∀ x ∈ Rn α ∈ R
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     ◮ ν(x +y) ≤ ν(x)+ν(y) ∀ x,y ∈ Rn
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   Weusually denote ν(x) by kxk. Norms are convex functions.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   l norms
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   p
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 P             1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    n       p p
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  kxk      := (         |x | )   ,  1 ≤ p < ∞
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      p             i=1   i
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  kxk       = max            |x |
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      ∞              i=1,...,n i
   Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem
   Multi-Variable Calculus
        Norms:
        Afunction ν : Rn → R is a vector norm on Rn if
   Multivariable Calculus Review
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     ◮ ν(αx) = |α|ν(x) ∀ x ∈ Rn α ∈ R
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     ◮ ν(x +y) ≤ ν(x)+ν(y) ∀ x,y ∈ Rn
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   Weusually denote ν(x) by kxk. Norms are convex functions.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   l norms
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   p
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 P             1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    n       p p
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  kxk      := (         |x | )   ,  1 ≤ p < ∞
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      p             i=1   i
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  kxk       = max            |x |
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      ∞              i=1,...,n i
   Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem
   Multi-Variable Calculus
        Norms:
        Afunction ν : Rn → R is a vector norm on Rn if
          ◮ ν(x) ≥ 0 ∀ x ∈ Rn with equality iff x = 0.
   Multivariable Calculus Review