Define the wedge of pointed spaces (X, x0),(Y, y0) ∈Top* to be (X ∨Y, z0), where X ∨Y is the quo

Define the wedge of pointed spaces (X, x0),(Y, y0) ∈Top* to...

Key Concepts

Pointed Topological Space

A pointed topological space is a topological space equipped with a distinguished element called a basepoint. This structure is essential for various constructions in algebraic topology, especially when one needs to specify a 'starting point' for homotopical or categorical constructions.

Quotient Topology

The quotient topology is the topology on a set obtained by partitioning a topological space and endowing the partition set with the finest topology that makes the natural projection map continuous. This concept is crucial for constructing new spaces from old ones by identifying certain subsets, such as the process of identifying basepoints in disjoint unions.

Wedge Sum

The wedge sum is a construction in topology where several pointed spaces are combined into a single space by identifying all their basepoints to a single point. This process creates a new space that retains the structure of each component while 'gluing' them together at a common point.

Coproduct in Category Theory

In category theory, the coproduct is a concept generalizing the disjoint union in topology or the direct sum in algebra. In the context of pointed spaces, it refers to the construction that satisfies a universal property, meaning that any function from the coproduct to another space corresponds uniquely to a collection of functions from each component that respect the basepoints.

Universal Property

The universal property is a defining condition in category theory that characterizes an object (or construction) up to unique isomorphism by the property of having a unique morphism from it to any other object that factors through a given family of maps. For the wedge sum in pointed spaces, the universal property ensures that any pair of basepoint-preserving maps from the individual spaces factors uniquely through the wedge, making it the coproduct in the category.

Transcript

00:02 We're given two subspaces, u and w, of a finite dimensional inner product space b.

00:25 And we're asked to prove properties about these subspaces.

00:34 So in part a, we're asked to prove that the orthogonal complement of u plus w is equal to the orthogonal complement of u intersected with the orthogonal complement of w.

00:49 So i'll begin with the left -hand side.

00:53 So let's, i'd say let u be a vector in the orthogonal complement of u plus w.

01:06 Actually, let's do v instead for clarity.

01:14 Then it follows that the inner product of v, let's call this, with x is equal to 0 for all vectors x in u plus w.

01:42 Now, since x is in u plus w, x is going to be equal to u plus w for some u in u and w in w, and therefore we have that the inner product of v with u plus w equals zero.

02:36 Sorry.

02:40 Take a different approach.

02:44 All right.

02:48 Instead, we'll let you be some vector in you.

02:50 We'll let you be some vector in you.

02:54 Then it follows that the vector x which is u plus zero well this is an element of u plus w and therefore it follows the inner product of v with x well this is the interproduct of v with u plus zero which is the inner product of v with u equals zero since this is true for any u it follows that v lies in the uh orthogonal complement of you.

03:31 Likewise, if w isn't w, then it follows that the vector x, which is 0 plus w, lies in u plus w.

03:47 And therefore, zero is the inner product of v with x, which is the interproduct of v with 0 plus w, which is the inner product of v with w.

03:58 And therefore v is also in the orthogonal complement of w...

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