Timestamps and Summary for Lecture 3: Quotient Spaces, the Baire Category Theorem and the Uniform Boundedness Theorem0:00 - Recap of Lecture 2Last lecture: linear operators between normed vector spaces are continuous if and only if they are bounded. The space of bounded operators between two such NVS's itself forms an NVS, under the operator norm. When the target is a Banach space, this space of bounded operators also becomes Banach: this is a way to identify new Banach spaces from existing ones, by considering all bounded operators into them. 0:59 - SubspacesA subspace of a given vector space is internally closed under addition and scaling (linear combinations). Theorem: a subspace W of a Banach space V is itself Banach if and only if it is a closed subspace of V with respect to the metric induced by the norm. Closure amounts to showing that W contains all of its limit points (every convergent sequence in W has its limit in W). We can take any sequence in W; by hypothesis (completeness) it is Cauchy and converges in W. Conversely, if the subspace is closed, we can take a Cauchy sequence in W -- regarding it as a sequence in the ambient Banach space V, it converges to an element in W by closure. This shows every Cauchy sequence in W converges in W, so W is Banach. 5:14 - Quotient SpacesAnother crucial way to form new spaces from old by passing to substructures involves the process of taking quotients. Given a subspace W of V, introduce (and verify) an equivalence relation on vectors in V by identifying two vectors together if their difference lies in W. This allows us to form the quotient set V/W under that equivalence relation; furthermore, it naturally inherits the operations from V and becomes a vector space in its own right. 11:31 - From Seminorms to NormsRecall that a seminorm satisfies homogeneity and the triangle inequality, but not necessarily definiteness -- consider the norm of the derivative of a function; it exhibits the former two properties but resolves to zero on constant functions. Theorem: a seminorm on a vector space descends to a norm on its quotient taken with the subspace of all vectors on which the seminorm is zero. Intuitively: the presence of vectors with seminorm zero represent an "obstruction" of the seminorm from becoming a norm. By collapsing the obstruction in the quotient, we obtain a true norm. 23:11 - Baire Category TheoremA cornerstone of general topology with vital applications in analysis. The version given here states that if a complete metric space can be expressed as the countable union of closed sets, then at least one of the closed sets contains an open ball (eq. has an interior point/nonempty interior). It can be reformulated to state that the countable intersection of open, dense sets in a complete metric space remains dense. (Note: a second result, not as relevant to functional analysis, states that the same holds true for locally compact regular spaces. This is especially important in the study of locally compact Hausdorff spaces. The version proven here requires some form of the Axiom of Choice; over ZF it turns out equivalent to a weaker form of Choice that is sufficient to build up most of real analysis). 50:39 - The Uniform Boundedness TheoremOur first major functional analysis result: it states that a sequence of pointwise-bounded operators on a Banach space is also bounded uniformly. More precisely, a sequence of linear operators from a Banach space to a normed vector space is bounded uniformly in the operator norm as soon as each individual operator is bounded. By showing that each of these operators have bounds that might vary with each point, this theorem claims that a bound can be chosen for all operators in the sequence that is independent of the point, and can be applied globally.####[]