Specifically, we will learn a vector space with some topology on it (called a topological vector space). Topological vector space (also called linear topological space) is one of the basic structures investigated by functional analysis. In Chapter 1 we will learn some separation properties in TVS (Topological Vector Space) in the sense that two disjoint closed and compact sets can be separated by finding suitable disjoint neighborhoods.
In Chapter 2, we will learn some definitions such as Banach space, dual of spaces, convex (affine) sets and convex (affine) hulls of sets, algebraic relative interior, algebraic closure of set and radial limit, etc. In Chapter 3, we will see Banach-Steinhaus theorem, open mapping theorem, and closed graph theorem.
Introduction
A subset E of a topological vector space is said to be bounded if for every neighborhood V of 0 in Associate each a ∈X and each scalar λ6= 0 with the translation operator Ta and the multiplication operator. Ta and Mλ are homeomorphisms of X on X. A local basis of a topological vector space The open sets of X are then precisely the sets of translations of members of B.
In the vector space context, the term local base will always mean a local base at 0. In the following definitions, X always denotes a topological vector space, with topology τ. a) X is locally convex if there is a local baseB whose members are convex.
Separation properties
If B is a local basis for a topological vector space X, then every member of B contains the closure of some member of B. A+V), where V passes through all neighborhoods of 0. a) every neighborhood of 0 contains an equilibrium neighborhood of 0, and (b) every convex neighborhood of 0 contains a balanced convex neighborhood. a) Every topological vector space has a balanced local basis.
Linear Mapping
Finite Dimensional Spaces
Metrization
Since B is a local zero basis that implies that V is a subset of every neighborhood of Y. If every Cauchy sequence in X converges to a point of X, then d is said to be a complete metric in X (say d- cauchy sequence). b) Let τ be the topology of a topological vector space X. Fix a local basis B for τ. If d1 and d2 are invariant metrics on a vector space X which induce the same topology on X, then.
Suppose Y is a subspace of the topological vector space X and Y is an F-space (in the topology inherited from X). By exactly the same argument, the Y-intersection of the closure of sets FnW contains exactly one point.
Boundedness and Continuity
T is said to be bounded if T maps bounded sets to bounded sets, i.e. if T(E) is a bounded subset of Y for every bounded set E ⊆X. As we stated, the space is metric, which means that it is a Hausdorff space, so sequential continuity means that T is connected.
Seminorms and Local Convexity
Assume that B is a convex balanced local basis in a topological vector space X. Associate each V ∈B with its Minkowski functional µV. b) {µV :V ∈B} is a separating family of continuous seminorms on X. Clearly, B is a collection of sets containing the origin and closed under finite intersection. If B is a convex balanced local basis of topology τ of locally convex space X, then B generates a separating family P of continuous semi-norm on X.
Let if X is normable and k.k is norm compatible with our topology on X then the open unit ball {x:kxk<1} is convex and bounded.
Quotient Space
If B is a local basis for τ, then the collection of all sets π(V) with V ∈B is a local basis for τN. Since π is open, π(V) is a neighborhood of 0 in X/N. Now let W be a neighborhood of 0 in X/N. Therefore, gauge multiplication is continuous.
Examples
Now let {fi} be the Cauchy sequence with respect to this metric d, an easy calculation shows that pn(fi −fj)→0, ∀n as i, j→. The spaces C∞(Ω) and DK : A complex function f defined in some nonempty open set Ω ⊂ Rn is said to belong to C∞(Ω) if Dαf ∈C(Ω) for every multi-index α. DK is the point of intersection between the kernels of these functions, where x ranges in Kc, therefore DK is closed in C∞(Ω).
Now the metric constructed from these seminorms which is compatible with the topology on C∞(Ω) is a bounded metric but no norm is bounded, so the metric is not induced by any norm. Every normed linear space has a natural topology such that (the metric and the metric induced by the norm induce a topology.) :. is continuous with this topology by the triangle inequality. is continuous with this topology by the triangle inequality and norm homogeneity. If f is a linear function in a linear topological space, then the following statements are equivalent: iii).
From now on, we consider X, Y to be normed linear space and K to be field (either R or C), unless stated. L(X, Y) The collection of all linear continuous map from X with values in Y which is a normed linear space by. A pre-Hilbert space is also considered a linear normed space by the norm induced by inner product.
Two elements x and y in pre-Hilbert space are said to be orthogonal if hx, yi= 0. If a pre-Hilbert space is complete in the norm associated with the given inner product, then it is called a Hilbert space. Riesz : If f is a continuous linear functional on the Hilbert space X, then there exists a unique element a∈X such that.
Convex Sets
Hence it is true for all finite n∈N. 1) The intersection of many arbitrary convex (affine) groups is again a convex (affine) group. Let C be any arbitrary subset of X, then the intersection of all convex (affine) sets containing C is called a convex (affine) hull, i.e. A point a0 ∈X (linear real space) is called a relative algebraic interior of A⊂X if, for every line through a0 extending to affA, there exists an open segment contained in A containing a0.
The set of all the algebraic (relative) interior points of A is called the algebraic (relative) interior of the set A, and we denote it by (Ari)Ai. The set of all relative interior points of A is called the relative interior of A and is denoted by riA. Let X be a finite-dimensional separated topological linear space and let A be a convex set of X.
A point x0 ∈A is algebraically interior of A if and only if x0 is an interior point (in the topological sense) of A. A point x0 ∈A, where A is a convex set from a finite-dimensional separated topological linear space, is an algebraic relative interior point of A if and only if it is a relative interior point of A. If A is a convex set where the origin is an algebraic relative interior point, then.
The interior of a convex set is either an empty set or coincides with its algebraic interior. A set is a hyperplane if and only if it is the translation of a maximal linear subspace. Conversely, for every homogeneous hyperplane H, there exists a functional, uniquely determined up to a non-zero multiplicative constant, with the kernel H.
Separation of Convex Sets
A hyperplane H is called a supporting hyperplane of a set A if H contains at least one point of A and A lies in one of the two closed half-spaces determined by H. If x0 ∈/ F, where F is a non-empty closed convex set of a separated locally convex space, then there exists a closed hyperplane that strictly separates F and x0, that is, there exists a nontrivial continuous linear functional such. The sets in the first category in S are those that are countable unions of nowhere dense sets.
Any subset S that is not of the first category is said to be of the second category in S. Z as a subspace of R is itself of the second category, but as Z⊂R it is of the first category. Therefore, complete metric spaces, as well as locally compact Hausdorff spaces, are inherently different categories.
Equicontinuity: Suppose that X and Y are topological vector spaces and Γ is a collection of linear maps from X to Y. We say that Γ is equicontinuous if for every quarter W of 0 in Y there corresponds a quarter V of 0 in X, such that T (V)⊂W for all T ∈Γ. Suppose that X and Y are topological vector spaces, Γ is an equicontinuous collection of linear maps from X to Y, and E is a bounded subset of X. Let W be a neighborhood of 0 in Y. Since Γ is equicontinuous, there is a quarter V of 0 in X s.t. Banach-Steinhaus): Suppose that X and Y are topological vector spaces, Γ is a collection of continuous linear mappings from X to Y, and B is the set of all x∈X whose orbits.
Now there is at least one nE of the second category X, as this holds for B. If Γ is the collection of continuous linear mappings from the F-space X to the topological vector space Y and if the sets. If Tn is a sequence of continuous linear mappings from the F-space X to the topological vector space Y and if.
The Open Mapping Theorem
The Closed Graph Theorem
If X, Y, Z are topological vector spaces and if every Bx and every By is connected, then B is said to be separately connected. Suppose that B : X × Y → Z is bilinear and separately connected, X is an F-space, and Y and Z are topological vector spaces. Suppose that M is a subspace of the vector space X, p is full-norm on X, and f is a linear functional on M, such that.
Weak Topologies
Suppose, without loss of generality, that |fn|61 for all n, and let τd be the topolgy induced by the metric on X. Since N ⊂V is infinite dimensional if X, every weak neighborhood of 0 contains an infinite dimensional subspace; hence Xw is not locally bounded. For convex subsets of a locally convex space, (1) originally closed is weakly closed and. 2) originally dense equals weakly dense.
If {xn} is a series in X that converges weakly to some x ∈ Since neighborhoods of 0 are absorbing, so corresponding to eachx∈X, there is numberγx <∞such that x∈γxV. Every originally bounded set is weakly bounded because every weak neighborhood of 0 in X is an original neighborhood of 0.
Since K is convex and weakly∗-compact and since the function T → T x is weakly∗-continuous, we conclude by Theorem 3.1.13 from equation (2) that there is a constant β <∞such. If E ⊂Rn andx∈co(E), then x lies in the convex hull of a subset of E containing at most n+ 1 points. 2) If X is a locally convex topological vector space and E ⊂ X is fully bounded, then co(E) is fully bounded. So if K is compact in a Frechet space, then K is clearly completely bounded; therefore co(K) is completely bounded by (2), and therefore co(K) is compact.
If K is a nonempty compact convex set in X, then K is the closed convex hull of the set of its extreme points. Sort P0 partially by set inclusion, let ω be a maximal total ordered subset of P0, and let M be the intersection of all members of ω. If K is a compact set in a locally convex space X, and if co(K) is also compact, then every extreme point of co(K) lies in K.
