Banach Spaces

A normed linear space is a vector space together with a norm function. The norm is like the concept of the length of an arrow. Then, the space admits a metric space structure, where the distance between two points is simply the length of the difference of the two points. Thus, we have all the tools for metric space and topological space at our disposal. A complete normed linear space is called a Banach space. For finite-dimensional Banach spaces, compactness of a set is equivalent to the set being closed and bounded. But, for infinite-dimensional Banach spaces, compactness is hard to come by. We present results on the quotient space of a Banach space with respect to a given subspace. Then, we switch gear and prove the Stone-Weierstrass Theorem 7.56. We present results on the linear operators, in particular, bounded linear operators defined between Banach spaces. Particular attention is spent with the dual of the original Banach space, where Hahn–Banach Theorems are proved. A Banach space is reflexive if it is isomorphically isometric to its second dual. Reflexive Banach space is simpler to deal with. Minimum norm problem from a vector to a subspace can be solved using the concept of orthogonal complement. The adjoint of a bounded linear operator A is A ′, whose properties are studied. In Propositions 7.112 and 7.114, the infinite-dimensional counterparts of the matrix fact dim ( ℛ A ) = n − dim ( N A ) $$\dim (\mathcal {R}{\left ( A \right ) }) = n - \dim (\mathcal {N}{\left ( A \right ) })$$ is presented. This chapter concludes with a discussion of Alaoglu Theorem 7.122 and the weak and weak∗ topologies.