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Definitions

Fionn´anHoward June 2011

Contents

1 1

2 Zorn’s Lemma 3

3 Nets and Separation Axioms 4

4 Banach Spaces I 5

5 Banach Spaces II 5

6 Main Theorems on Banach Spaces 6

7 Hilbert Spaces 6

1 1 Topology

• -A is a X with a d : X × X → [0, ∞) such that for all x, y, z ∈ X:

1. d(x, y) ≥ 0 with equality iff x = y 2. d(x, y) = d(y, x) 3. d(x, z) ≤ d(x, y) + d(y, z)

• Open - The open ball of radius r and centre x0 is given by,

B(x0, r) = {x ∈ X : d(x, x0) < r}.

• Closed Ball - The closed ball of radius r and centre x0 is given by,

B¯(x0, r) = {x ∈ X : d(x, x0) ≤ r}.

• / - For a G ⊆ X of a metric space X, a point x ∈ G is an interior point of G if ∃ r > 0 such that B(x, r) ⊂ G. The set G is called an open set if every x ∈ G is an interior point. • - A set G ⊆ X of a metric space X is called a closed set if it’s X\G is an open set.

• - For a subset T ⊂ X of a (X, τ), we define the subspace topology τT on T to be, τT = {T ∩ V : V ∈ τ}.

• Interior / / - The interior Eo of a set E is the of all open sets contained in E. The closure E¯ of a set E is the intersection of all closed sets containing E. The boundary ∂E of a set E is given by E¯\Eo.

• Continuity in a Metric Space - For metric spaces (X, dX ) and (Y, dY ), a function f : X → Y is called continuous at x0 ∈ X if for each  > 0, ∃ δ > 0 such that,

dX (x, x0) < δ =⇒ dY (f(x), f(x0)) < .

• Continuity in a Topological Space - For topological spaces (X, τX ) and (Y, τY ), a function f : X → Y is called continuous if: −1 f (V ) ∈ τX for all V ∈ τY .

∞ • in a Metric Space - For a metric space (X, d), a (xn)n=1 ∈ X converges to a limit l ∈ X (denoted lim xn = l) if for each  > 0, ∃ N ∈ such that, n→∞ N

n ∈ N, n > N =⇒ d(xn, l) < .

2 ∞ • Limit in a Topological Space - For a topological space (X, τ), a sequence (xn)n=1 ∈ X converges to a limit l ∈ X if for each open set U with l ∈ U, ∃ N ≥ 1 such that,

n ∈ N, n > N =⇒ xn ∈ U.

• Open / Compact - For a topological space (X, τ) and a subset T ⊆ X, an open cover of T is a family U of open of X such that,

T ⊆ ∪{U : U ∈ U}.

T is compact if each open cover has a finite subcover (subfamily of U which is also a cover of T ).

∞ • Sequentially Compact - A subset T ⊂ X is called sequentially compact if each sequence (tn)n=1 has ∞ a subsequence (tnj )j=1 that converges to some limit l ∈ X. • Connected - A topological space X is called connected if the only clopen subsets of X are ∅ and X itself. For subsets T ⊂ X take T with the subspace topology, (T, τT ). • - A subfamily B ⊆ τ is called a base (for the open sets) of the topology if for each x ∈ U open, there exists B ∈ B such that x ∈ B ⊂ U. • Second Countable - second countable if ∃ a countable base. • Dense - A subset S ⊆ X is said to be dense in X if S¯ = X. • Separable - separable if ∃ countable dense subset S of X.

• Weaker/Stronger - Let X be a set with two topologies τ1 and τ2. τ1 is weaker than τ2 if τ1 ⊆ τ2, i.e. there are fewer open sets in the weaker topology. • - Given two top. spaces X and Y we define the product topology to be the topology which has as its base, {U × V : U ⊂ X open,V ⊂ Y open}.

◦ • Neighbourhood - A subset N ⊆ X is a nbd of x0 if x0 ∈ N .

• - The nbd system Ux0 at x0 is the collection of all nbds of x0.

• Neighbourhood Base -A nbd base at a point x0 ∈ X is a subset Bx0 ⊂ Ux0 such that for each

N ∈ Ux0 , there exists B ∈ Bx0 with B ⊆ N.

• First Countable -(X, τ) is first countable at x0 ∈ X if ∃ a countable nbd base at x0.

2 Zorn’s Lemma

• Partial Order - The relation ≤ is a partial order on S if for all x, y, z ∈ S: 1. x ≤ x. 2. x ≤ y and y ≤ x =⇒ x = y. 3. x ≤ y and y ≤ z =⇒ x ≤ z .

3 • Linear Ordering -A linear ordering ≤ on a set S is a partial order with the additional property that for all x, y ∈ S, either x ≤ y or y ≤ x. • Chain - For a poset (S, ≤), a chain in S is a subset C ⊆ S that is linearly ordered. • Upper Bound - For a poset (S, ≤), an elt u ∈ S is an upper bound for R ⊆ S if for all x ∈ R, x ≤ u. • Maximal Element - For a poset (S, ≤), m ∈ S is a maximal element of S if for x ∈ S with m ≤ x, x = m. • Linearly Independent - If V is a v.s. over a field K, a subset E ⊆ V is linearly independent if for distinct e1, . . . , en ∈ E and λ1, . . . , λn ∈ K, we have,

λ1e1, . . . , λnen = 0 =⇒ λ1 = ... = λn = 0.

Span - E spans V if all v ∈ V can be expressed as v = λ1e1, . . . , λnen. - E is a basis for V if it is linearly independent and spans V .

3 Nets and Separation Axioms

• Hausdorff Space -(X, τ) is called a Hausdorff (or T2) space if for a, b ∈ X with a 6= b, there exists U, V ⊂ X with a ∈ U, b ∈ V and U ∩ V = ∅. • (Λ, ≤) is a directed set if: 1. λ ≤ λ

2. λ1 ≤ λ2 and λ2 ≤ λ3 implies λ1 ≤ λ3

3. For all λ1, λ2 ∈ Λ there exists λ3 ∈ Λ with λ1 ≤ λ3 and λ2 ≤ λ3.

• -A net (xλ)λ∈Λ in a set X (usually a top. space) consists of: 1. a directed set (Λ, ≤) 2. a function x :Λ → X

• Converges - A net converges to x0 ∈ X if for each U open with x0 ∈ U, there exists λ0 ∈ Λ such that for λ ≥ λ0, xλ ∈ U. • Cofinal - A subset A ⊆ Λ is called cofinal in Λ if for each λ ∈ Λ there exists α ∈ A with λ ≤ α.

• - For a net (xλ)λ∈Λ indexed by a directed set (Λ, ≤Λ), a subnet consists of two things:

1. another net (yα)α∈A indexed by a set (A, ≤A) 2. a map j : A → Λ such that

(a) j is order preserving, α1 ≤A α2 =⇒ j(α1) ≤Λ j(α2)

(b) yα = xj(α) (c) j(A) is cofinal in Λ.

• Cluster/ - A point x0 ∈ X is called a cluster/limit point of a net if for N ∈ Ux0 , the set {λ ∈ Λ: xλ ∈ N} is cofinal in Λ.

4 • T0 Space - if ∃ U ⊆ X open with either x ∈ U, y∈ / U or y ∈ U, x∈ / U.

• - if ∃ U ⊆ X open such that x ∈ U, y∈ / U.

• Regular/T3 Space - if A ⊆ X closed, and b ∈ X\A implies ∃ U, V ⊆ X open such that A ⊆ U, b ∈ V and U ∩ V = ∅.

T3 if it is both regular and Hausdorff.

• Normal/T4 Space - if A, B ⊆ X closed, A ∩ B = ∅ implies ∃ U, V ⊆ X open such that A ⊆ U, B ⊆ V an U ∩ V = ∅.

T4 if it is both normal and Hausdorff.

• Completely Regular/T 1 Space - A topological space is called completely regular if for A ⊂ X 3 2 closed with b ∈ X\A, there exists continuous f : X → R with f(a) = 0 for all a ∈ A and f(b) = 1.

T 1 if it is both completely regular and Hausdorff. 3 2

4 Banach Spaces I

• -A norm on a v.s. E over K is a function kxk : E → [0, ∞) such that for all x, y ∈ E: 1. kx + yk ≤ kxk + kyk 2. kλxk = |λ|kxk for all λ ∈ K 3. kxk = 0 =⇒ x = 0

• -A seminorm (denoted p(x) instead of kxk) is a norm that doesn’t satisfy 3 above. Non-zero elts can have length 0.

∞ • - Note that this definition requires a metric. A sequence (xn)n=1 in a metric space is called a Cauchy sequence if for each  > 0, ∃N such that,

n, m ≥ N =⇒ d(xn, xm) < .

• Complete - A metric space is called complete if every Cauchy sequence converges to a limit. • Uniformly Continuous - A function f : X → Y is called uniformly continuous if for each  > 0, ∃ δ > 0 such that, d(x1, x2) < δ =⇒ d(f(x1), f(x2)) < .

• - A normed space (E, k · k) over K is called a Banach space over K if E is complete in the metric arising from the norm.

5 5 Banach Spaces II

• (Bounded) Linear Operator -A bounded linear operator is a linear transformation T : E → F between normed spaces such that for all x ∈ E, kT xkF ≤ MkxkE for some M ≥ 0. • - If T : E → F is a bounded linear operator between normed spaces, then the operator norm of T is given by,

kT kop = inf{M ≥ 0 : kT xkF ≤ MkxkE for all x ∈ E}.

• Dual Space - The dual space E∗ of a normed space E is given by,

∗ E = B(E, K) = {T : E → K | T continuous and linear}. Elements of E∗ are called linear functionals on E. We use the operator norm on E∗.

• Isomorphic - Two normed spaces (E, k · kE) and (F, k · kF ) are isomorphic if there exists a v.s. isomorphism T : E → F which is also a .

• Isometrically Isomorphic - Two normed spaces (E, k·kE) and (F, k·kF ) are isometrically isomorphic if there exists a v.s. isomorphism T : E → F which is also isometric (kT xkF = kxkE for all x ∈ E).

6 Main Theorems on Banach Spaces

• Nowhere Dense - A subset S of a metric or topological space is called nowhere dense if (S¯)◦ = ∅. • First - A subset E ⊂ X is said to be of first category if it is a countable union of nowhere dense sets. • Second Category - If its not first category...obviously! • / Partial Sum in a Normed Space - In a normed space (E, k · k), a series in E is a sequence ∞ (xn)n=1 where xn ∈ E. The partial sum of the series is,

n X sn = xj. j=1 We say that the series converges in E if the sequence of partial sums has a limit,   n X lim  xj − s = 0. n→∞ j=1

∞ ∞ X X • Absolutely Convergent - The series xn is absolutely convergent if kxnk < ∞. Note that this n=1 n=1 is a sequence of positive terms and so as monotone increasing sequence of partial sums. So it either converges in R or increases to ∞. • Reflexive Space -

6 7 Hilbert Spaces

• Inner Product - An inner product is a map h·, ·i : V × V → K on a v.s. V over K such that: 1. hx + y, zi = hx, zi + hy, zi 2. hλx, yi = λ hx, yi 3. hx, yi = hy,¯ xi 4. hx, xi ≥ 0 with equality iff x = 0 It also follows that

– hx, y + zi = hx, yi + hx, zi – hx, λyi = λ¯ hx, yi • -A Hilbert space is an that is complete in the norm kxk = phx, xi arising from the inner product (i.e. complete in the metric arising from the norm). In other words, its a Banach space with this norm.

• Orthogonal - Two elts x, y of an inner-product space are called orthogonal if hx, yi = 0. A subset S ⊂ H of Hilbert space (or inner product space) is called orthogonal if hx, yi = 0 for all x, y ∈ S, x 6= y. • Orthonormal - S is called orthonormal if it is orthogonal and kxk = 1 for all x ∈ S.

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