View source for Bounded operator

A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b0174201.png" /> of a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b0174202.png" /> into a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b0174203.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b0174204.png" /> is a bounded subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b0174205.png" /> for any bounded subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b0174206.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b0174207.png" />. Every operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b0174208.png" />, continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b0174209.png" />, is a bounded operator. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742010.png" /> is a linear operator, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742011.png" /> to be bounded it is sufficient that there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742013.png" /> is bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742014.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742016.png" /> are normed linear spaces and that the linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742017.png" /> is bounded. Then

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This number is called the norm of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742019.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742020.png" />. Then

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and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742022.png" /> is the smallest constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742023.png" /> such that

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for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742025.png" />. Conversely, if this inequality is satisfied, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742026.png" /> is bounded. For linear operators mapping a normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742027.png" /> into a normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742028.png" />, the concepts of boundedness and continuity are equivalent. This is not the case for arbitrary topological vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742030.png" />, but if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742031.png" /> is bornological and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742032.png" /> is a locally convex space, then the boundedness of a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742033.png" /> implies its continuity. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742034.png" /> is a Hilbert space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742035.png" /> is a bounded symmetric operator, then the quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742036.png" /> is bounded on the unit ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742037.png" />. The numbers

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are called the upper and lower bounds of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742039.png" />. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742041.png" /> belongs to the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742042.png" />, and the whole spectrum lies in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742043.png" />. Examples of bounded operators are: the projection operator ([[Projector|projector]]) onto a complemented subspace of a Banach space, and an [[Isometric operator|isometric operator]] acting on a Hilbert space.

If the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742045.png" /> have the structure of a partially ordered set, for example are vector lattices (cf. [[Vector lattice|Vector lattice]]), then a concept of order-boundedness of an operator can be introduced, besides the topological boundedness considered above. An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742046.png" /> is called order-bounded if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742047.png" /> is an order-bounded set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742048.png" /> for any order-bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742049.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742050.png" />. Examples: an isotone operator, i.e. an operator such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742051.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742052.png" />.

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



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.E. Taylor,   D.C. Lay,   "Introduction to functional analysis" , Wiley  (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.C. Zaanen,   "Riesz spaces" , '''II''' , North-Holland  (1983)</TD></TR></table>