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 I ν(x) ≥ 0 8 x 2 R with equality iff x = 0. n I ν(αx) = jαjν(x) 8 x 2 R α 2 R n I ν(x + y) ≤ ν(x) + ν(y) 8 x; y 2 R We usually denote ν(x) by kxk. Norms are convex functions. lp norms 1 Pn p p kxkp := ( i=1 jxi j ) ; 1 ≤ p < 1 kxk1 = maxi=1;:::;n jxi j Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Multi-Variable Calculus Norms: n n A function ν : R ! R is a vector norm on R if Multivariable Calculus Review n I ν(αx) = jαjν(x) 8 x 2 R α 2 R n I ν(x + y) ≤ ν(x) + ν(y) 8 x; y 2 R We usually denote ν(x) by kxk. Norms are convex functions. lp norms 1 Pn p p kxkp := ( i=1 jxi j ) ; 1 ≤ p < 1 kxk1 = maxi=1;:::;n jxi j Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Multi-Variable Calculus Norms: n n A function ν : R ! R is a vector norm on R if n I ν(x) ≥ 0 8 x 2 R with equality iff x = 0. Multivariable Calculus Review We usually denote ν(x) by kxk. Norms are convex functions. lp norms 1 Pn p p kxkp := ( i=1 jxi j ) ; 1 ≤ p < 1 kxk1 = maxi=1;:::;n jxi j Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Multi-Variable Calculus Norms: n n A function ν : R ! R is a vector norm on R if n I ν(x) ≥ 0 8 x 2 R with equality iff x = 0. n I ν(αx) = jαjν(x) 8 x 2 R α 2 R n I ν(x + y) ≤ ν(x) + ν(y) 8 x; y 2 R Multivariable Calculus Review lp norms 1 Pn p p kxkp := ( i=1 jxi j ) ; 1 ≤ p < 1 kxk1 = maxi=1;:::;n jxi j Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Multi-Variable Calculus Norms: n n A function ν : R ! R is a vector norm on R if n I ν(x) ≥ 0 8 x 2 R with equality iff x = 0. n I ν(αx) = jαjν(x) 8 x 2 R α 2 R n I ν(x + y) ≤ ν(x) + ν(y) 8 x; y 2 R We usually denote ν(x) by kxk. Norms are convex functions. 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 Norms: n n A function ν : R ! R is a vector norm on R if n I ν(x) ≥ 0 8 x 2 R with equality iff x = 0. n I ν(αx) = jαjν(x) 8 x 2 R α 2 R n I ν(x + y) ≤ ν(x) + ν(y) 8 x; y 2 R We usually denote ν(x) by kxk. Norms are convex functions. lp norms 1 Pn p p kxkp := ( i=1 jxi j ) ; 1 ≤ p < 1 kxk1 = maxi=1;:::;n jxi j Multivariable Calculus Review I D is open if for every x 2 D there exists > 0 such that x + B ⊂ D where x + B = fx + u : u 2 Bg n and B is the unit ball of some given norm on R . I D is closed if every point x satisfying (x + B) \ D 6= ; for all > 0, must be a point in D. I D is bounded if there exists β > 0 such that kxk ≤ β for all x 2 D: Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Elementary Topological Properties of Sets n Let D ⊂ R . Multivariable Calculus Review I D is closed if every point x satisfying (x + B) \ D 6= ; for all > 0, must be a point in D. I D is bounded if there exists β > 0 such that kxk ≤ β for all x 2 D: Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Elementary Topological Properties of Sets n Let D ⊂ R . I D is open if for every x 2 D there exists > 0 such that x + B ⊂ D where x + B = fx + u : u 2 Bg n and B is the unit ball of some given norm on R . Multivariable Calculus Review I D is bounded if there exists β > 0 such that kxk ≤ β for all x 2 D: Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Elementary Topological Properties of Sets n Let D ⊂ R . I D is open if for every x 2 D there exists > 0 such that x + B ⊂ D where x + B = fx + u : u 2 Bg n and B is the unit ball of some given norm on R . I D is closed if every point x satisfying (x + B) \ D 6= ; for all > 0, must be a point in D. Multivariable Calculus Review Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Elementary Topological Properties of Sets n Let D ⊂ R . I D is open if for every x 2 D there exists > 0 such that x + B ⊂ D where x + B = fx + u : u 2 Bg n and B is the unit ball of some given norm on R . I D is closed if every point x satisfying (x + B) \ D 6= ; for all > 0, must be a point in D. I D is bounded if there exists β > 0 such that kxk ≤ β for all x 2 D: Multivariable Calculus Review I x 2 D is said to be a cluster point of D if (x + B) \ D 6= ; for every > 0. I D is compact if it is closed and bounded. Theorem: [Weierstrass Compactness Theorem] n A set D ⊂ R is compact if and only if every infinite subset of D has a cluster point and all such cluster points lie in D. Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Elementary Topological Properties of Sets: Compactness n Let D ⊂ R . Multivariable Calculus Review I D is compact if it is closed and bounded. Theorem: [Weierstrass Compactness Theorem] n A set D ⊂ R is compact if and only if every infinite subset of D has a cluster point and all such cluster points lie in D. Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Elementary Topological Properties of Sets: Compactness n Let D ⊂ R . I x 2 D is said to be a cluster point of D if (x + B) \ D 6= ; for every > 0. Multivariable Calculus Review Theorem: [Weierstrass Compactness Theorem] n A set D ⊂ R is compact if and only if every infinite subset of D has a cluster point and all such cluster points lie in D. Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Elementary Topological Properties of Sets: Compactness n Let D ⊂ R . I x 2 D is said to be a cluster point of D if (x + B) \ D 6= ; for every > 0. I D is compact if it is closed and bounded. Multivariable Calculus Review Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Elementary Topological Properties of Sets: Compactness n Let D ⊂ R . I x 2 D is said to be a cluster point of D if (x + B) \ D 6= ; for every > 0. I D is compact if it is closed and bounded. Theorem: [Weierstrass Compactness Theorem] n A set D ⊂ R is compact if and only if every infinite subset of D has a cluster point and all such cluster points lie in D. Multivariable Calculus Review n F is continuous on a set D ⊂ R if F is continuous at every point of D. Theorem: [Weierstrass Extreme Value Theorem] Every continuous function on a compact set attains its extreme values on that set. Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Continuity and The Weierstrass Extreme Value Theorem n m The mapping F : R ! R is continuous at the point x if lim kF (x) − F (x)k = 0: kx−xk!0 Multivariable Calculus Review Theorem: [Weierstrass Extreme Value Theorem] Every continuous function on a compact set attains its extreme values on that set. Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Continuity and The Weierstrass Extreme Value Theorem n m The mapping F : R ! R is continuous at the point x if lim kF (x) − F (x)k = 0: kx−xk!0 n F is continuous on a set D ⊂ R if F is continuous at every point of D.