Abstract: A Banach space is a complete normed vector space. A Hilbert space is a Banach space whose norm is induced by an inner product. Historically, many of the most important questions in the study of Banach spaces, such as "Does every separable Banach space have a basis?", are completely trivial for Hilbert spaces. Thus many problems in the area can be thought of as trying to understand if well known principles in Hilbert spaces can be extended to the general Banach space setting. On the other hand, the invariant subspace problem, which has been solved for certain Banach spaces, remains wide open for a separable Hilbert space. We will discuss some of the history of Banach spaces as well as recent results using the context of Hilbert spaces as motivation. The recent results will focus on frames for Banach spaces and the scalar plus compact problem.
