et
and
be Banach spaces and let
be a Hilbert space.

Definition

The mapping
is a "linear operator"
if

The "range of
"
is the
set
The "null space of
"
is the
set

A linear operator is "bounded"
if

A linear operator
is "adjoint" to
if

An operator
is "compact" if for any bounded sequence
the sequence
has a convergent subsequence.

For
,
the "resolvent set" of
is the
set
The "spectrum"
of
is the complement of
.
The "point spectrum"
is
The
is called "eigenvalue" and
consists of "eigenvectors".

Proposition

If
is compact then
is compact.

Proposition

(Fredholm alternative). Let
be a compact linear operator in Hilbert space
.
Then