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Ramsey’s theorem (in its infinite version) states that given any $k$-coloring $f:[\omega]^n \to k$ of a countably infinite set say, $\omega$, there is an infinite subset $A \subseteq \omega$ such that $f([A]^n)$ is constant, in which case we say that $f$ is homogeneous on $A$. In other words, for any coloring of the edges of a graph with countable vertices, there is some infinite subset of vertices whose edges are all of the same color.
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The Spectral Theorem comes in many flavors, one might encounter it first in it’s finite dimensional case, for $\mathbb{C}^n$. If $T:\mathbb{C}^n \to \mathbb{C}^n$ is a self-adjoint linear map then there is a basis ${x_1, …, x_n}$ of $\mathbb{C}^n$ consisting of eigenvectors of $T$ with which we have: \(Tx= \sum_{i=1}^n \lambda_i \langle x, x_i\rangle x_i.\) As its name suggests, it falls to Spectral Theory to extend this result into infinite dimensional spaces. As it’s usually the case, this brings forth considerable difficulties. One way of easing these difficulties is to retain some notion of finitude, in this case compactness.
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The spectrum $\sigma(T)$ of operators $T$ helps us them, and many of the problems spectral theorists face involve restraining this set $\sigma(T)$.
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One famous fact in functional analysis is that the closed unit ball $B_X$ of a normed space $X$ is compact if, and only if, the space $X$ is finite dimensional. As if paying homage to that, compact linear operators (operators that map bounded sets $A$ to relatively compact sets $T(A)$, where $\overline{T(A)}$ is compact), allude to both notions of compactness and finite-dimensionality.
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Neumann series are series of bounded linear operators of the form
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Essentially, spectral theory seeks to further the study of eigenvalues for the case of linear operators $T:X \to X$ on infinite dimensional spaces $X$.