Chapter III

Compact linear mappings and perturbations

In this chapter, all the vector spaces considered are vector spaces over a field K equal to $\mathbf{R}$ or $\mathbf{C}$. The references to EVT, II made in this book concern in general the case where $K =\mathbf{R}$; for the case where $K =\mathbf{C}$, cf. EVT, II, p. 64 to 68.

A semi-normed space is called a vector space E endowed with a semi-norm $p$ and the topology defined by $p($cf. EVT, II, p. 2). The unit ball of E, or of $p$, is called the set of elements $x$ of E such that $p(x)\leqslant 1$.

For every vector space E, $1_E$ denotes the identity mapping of E. If E, F and G are vector spaces and $u: E\rightarrow F,v: F\rightarrow G$ are linear mappings, one sometimes denotes by $vu$ the linear mapping $v\circ u$ from E into G.

Let E and F be topological vector spaces. A continuous linear mapping from E into F will also be called a morphism and an endomorphism when F = E. A bijective linear mapping from E into F which is continuous as well as its inverse will be called an isomorphism and, when F = E, an automorphism. When E and F are semi-normed spaces, the notions of endomorphism, isomorphism and automorphism refer to the underlying topological vector space structure.

Given topological vector spaces E and F, $\mathscr{L}^f(E; F)$ denotes the vector space of continuous linear mappings of finite rank from E into F. One also denotes by $\mathscr{L}^f(E)$ the vector space $\mathscr{L}^f(E; E)$.

Recall that a continuous linear mapping $u$ from E into F is strict if it induces by passing to the quotient an isomorphism of $E/$ Ker($u$) into $u(E)$ (TG, III, p. 16, déf. 1); it is equivalent to saying that the image of every neighbourhood of 0 in E is a neighbourhood of 0 in $u(E)$ (TG, III, p. 16, prop. 24).