# Bounded operator

A mapping $A$ of a topological vector space $X$ into a topological vector space $Y$ such that $A (M)$ is a bounded subset in $Y$ for any bounded subset $M$ of $X$. Every operator $A : X \rightarrow Y$, continuous on $X$, is a bounded operator. If $A : X \rightarrow Y$ is a linear operator, then for $A$ to be bounded it is sufficient that there exists a neighbourhood $U \subset X$ such that $A (U)$ is bounded in $Y$. Suppose that $X$ and $Y$ are normed linear spaces and that the linear operator $A : X \rightarrow Y$ is bounded. Then

$$\gamma = \ \sup _ {\| x \| \leq 1 } \| A x \| < \infty .$$

This number is called the norm of the operator $A$ and is denoted by $\| A \|$. Then

$$\| A x \| \leq \| A \| \cdot \| x \| ,$$

and $\| A \|$ is the smallest constant $C$ such that

$$\| A x \| \leq C \| x \|$$

for any $x \in X$. Conversely, if this inequality is satisfied, then $A$ is bounded. For linear operators mapping a normed space $X$ into a normed space $Y$, the concepts of boundedness and continuity are equivalent. This is not the case for arbitrary topological vector spaces $X$ and $Y$, but if $X$ is bornological and $Y$ is a locally convex space, then the boundedness of a linear operator $A : X \rightarrow Y$ implies its continuity. If $H$ is a Hilbert space and $A : H \rightarrow H$ is a bounded symmetric operator, then the quadratic form $\langle A x , x \rangle$ is bounded on the unit ball $K _ {1} = \{ {x } : {\| x \| \leq 1 } \}$. The numbers

$$\beta = \ \sup _ {\| x \| \leq 1 } \langle A x , x \rangle \ \ \textrm{ and } \ \ \alpha = \inf _ {\| x \| \leq 1 } \langle A x , x \rangle$$

are called the upper and lower bounds of the operator $A$. The points $\alpha$ and $\beta$ belongs to the spectrum of $A$, and the whole spectrum lies in the interval $[ \alpha , \beta ]$. Examples of bounded operators are: the projection operator (projector) onto a complemented subspace of a Banach space, and an isometric operator acting on a Hilbert space.

If the space $X$ and $Y$ have the structure of a partially ordered set, for example are vector lattices (cf. Vector lattice), then a concept of order-boundedness of an operator can be introduced, besides the topological boundedness considered above. An operator $A : X \rightarrow Y$ is called order-bounded if $A (M)$ is an order-bounded set in $Y$ for any order-bounded set $M$ in $X$. Examples: an isotone operator, i.e. an operator such that $x \leq y$ implies $A x \leq A y$.

#### References

 [1] L.A. Lyusternik, V.I. Sobolev, "Elemente der Funktionalanalysis" , Akademie Verlag (1955) (Translated from Russian) [2] W. Rudin, "Functional analysis" , McGraw-Hill (1973) [3] G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1967)