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B4a Banach spaces Michaelmas 2010 lectured by Bernd Kirchheim based on notes by CJK Batty
Lectures 7 and 8
- Theorem Let X be a normed vector space and let S be a countable subset of X such that span(S) = X 1 , then X is separable.
- Corollary Let X be a normed vector space and S ⊂ X. Then
spanS = spanS.
Hence, if there exists a countable set S ⊂ X such that spanS = X, then X is separable.
Remark Beside telling us when a space is separable, the Corollary also gives us a tool how to show that a certain linear subspace is dense - in the Example 6.(iii) below we see that “nice” functions (continuous and vanishing “near infinity”) are dense in the space of all (possibly very wild) Lebesgue integrable functions.
- Examples (i) The space
Ca,b = {f : R → F : f continuous and f (x) = 0 ∀x /∈ [a, b]}
with the supremum norm is separable. This follows easily using piecewise affine approximations with rational values in the divisionpoints. The crucial argument is that uniform continuity of an f ∈ C[a,b] ensures that the pwa approximations get uniformly close to f if the division is fine enough. (ii) Cc(R) = {f : R → R : f continuous & ∃Rf : |x| > Rf ⇒ f (x) = 0}, the space of all continuous functions vanishing near infinity ( also said to have a compact support ), with the supremums norm is separable. (iii) Cc(R) =
n=1 C[−n,n]^ is dense in (L
(^1) (R), ‖ · ‖ 1 ), and hence L (^1) (R) is
separable.
(^1) span(S) is the smallest linear subspace of X containing S
We consider the set C = {χM : M ⊂ R bounded and measurable}, then span(C) contains all simple functions, by a4 integration these are dense in L^1. We show that C ⊂ Cc(R) in L^1 and than use Corollary 5. To establish this claim
- for each U open bounded construct continuous : fn : R → [0, 1], fn ≤ χU and fn ր χU. Then Lebesgue’s DCT implies that fn → χU in L^1 so we conclude χU ∈ Cc(R)
- for each M bounded measurable and ε > 0 find U ⊃ M open bounded s.t. m(U \ M ) < ε. Hence
dist(χM , Cc(R)) ≤ ‖χM − χU ‖ 1 = ‖χU \M ‖ 1 < ε,
so sending ε ց 0 we have χM ∈ Cc(R).
5 Linear Operators
- Definition Let X, Y be normed vector spaces. A mapping T : X → Y such that for all x 1 , x 2 ∈ X and α 1 , α 2 ∈ C
T (α 1 x 1 + α 2 x 2 ) = α 1 T x 1 + α 2 T x 2
is said to be a Linear Operator from X to Y.
The linear operator T is said to be bounded if there exists M > 0 such that for all x ∈ X ‖T x‖Y ≤ M ‖x‖X
Let B(X, Y ) denote the set of bounded linear operators from X to Y (we write B(X) = B(X, Y ) if X = Y ).
- Lemma Let X, Y be normed vector spaces, let T ∈ B(X, Y ) and let
N 1 (T ) = inf {M : ‖T x‖ ≤ M ‖x‖ ∀x ∈ X} N 2 (T ) = sup{‖T x‖ : x ∈ B¯ 1 } ( B¯ 1 = {x ∈ X : ‖x‖ ≤ 1 })
N 3 (T ) = sup{
‖T x‖ ‖x‖
: x ∈ X and x 6 = 0}
N 4 (T ) = sup{‖T x‖ : x ∈ X ‖x‖ = 1}
