hi.(adsbygoogle = window.adsbygoogle || []).push({});

i'm reading "quantum mechanics in hilbert space" and a don't get a basic point for bounded operators.

def. 1 a set S in a normed space [tex]N[/tex] is bounded if there is a constant C such that [tex]\left\| f \right\| \leq C ~~~~~ \forall f \in S[/tex]

def. 2 a transformation is called bounded if it maps each bounded set into a bounded set.

and now comes the part i don't understand.

for linear operators [tex]T: N_1 \rightarrow N_2[/tex] def. 2 is equivalent to:

there exists a constant C such that [tex]\left\| T f \right\| \leq C \left\| f \right\| ~~~~~ \forall f \in N_1[/tex]

this is stated without a proof. i don't think it's obvious or at least not to me.

i'm thinking of a map, for example from the real numbers (normed space) to the real numbers, where the bounded set [tex]N_1=(0,1][/tex] is transformed in a way to another interval say [tex]N_2=(a,b][/tex] now the norm of elements from [tex]N_1[/tex] can get arbitrary small. So there can't exist a constant fulfilling [tex]\left\| T f \right\| \leq C \left\| f \right\| ~~~~~ \forall f \in N_1[/tex] when the norm of all elements of [tex]N_2[/tex] has a lower bound say [tex]m>0[/tex].

Or is such a map forbidden because of the continuity (zero has to be mapped on zero) and bounded linerar operators are continuous and vice verca?

i would be glad if someone can show me a proof or a source where a can get one.

thanks and greetings.

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Bounded Operators

Loading...

Similar Threads - Bounded Operators | Date |
---|---|

A On spectra | Feb 21, 2018 |

Spectra of T and T* when T is a bounded linear operator | Jan 14, 2011 |

The determinant of the bounded operator | Nov 1, 2010 |

**Physics Forums - The Fusion of Science and Community**