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Extra resources for A Course in Mathematical Analysis, vol. 2: Metric and Topological Spaces, Functions of a Vector Variable
Sample text
Suppose that (xn )∞ n=1 is a sequence in a metric space (X, d), and that x ∈ X. Set f (n) = xn , f (+∞) = x. Show that xn → x as n → ∞ if and only if f : (N, ρ) → (X, d) is continuous. 3 Suppose that (X, d), (Y, ρ) and (Z, σ) are metric spaces, that f is a continuous surjective mapping of (X, d) onto (Y, ρ) and that g : (Y, ρ) → (Z, σ) is continuous. Show that if g ◦ f is a homeomorphism of (X, d) onto (Z, σ) then f is a homeomorphism of (X, d) onto (Y, ρ) and g is a homeomorphism of (Y, ρ) onto (Z, σ).
X ∈ ∂A if and only if every open -neighbourhood of x contains an element of A and an element of C(A). A metric space is separable if it has a countable dense subset. Thus R, with its usual metric, is a separable metric space. 13 If (X, d) is a metric space with at least two points and if S is an infinite set, then the space BX (S) of bounded mappings from S → X, with the uniform metric, is not separable. 10. Suppose that x0 and x1 are distinct points of X, and let d = d(x0 , x1 ). For each subset A of X, define the mapping fA : S → X by setting fA (s) = x1 if s ∈ A and fA (s) = x0 if x ∈ A.
Since W ∩ W ⊥ = {0}, it follows that V = W ⊕ W ⊥ . If x ∈ V we can write x uniquely as y + z, with y ∈ W and z ∈ W ⊥ . P us set PW (x) = y. PW is a linear mapping of V onto W , and PW W W is called the orthogonal projection of V onto W . Note that PW ⊥ = I − PW . Although it is easy, the next result is important. It shows that an orthogonal projection is a ‘nearest point’ mapping; since it is linear, it relates the linear structure to metric properties. 3 If W is a linear subspace of a Euclidean or unitary space V and x ∈ V then PW (x) is the nearest point in W to x, and is the unique point in W with this property: x − PW (x) ≤ x − w for w ∈ W , and if x − PW (x) = x − w then w = PW (x).