basic properties of seminorms
Proposition 1.
Suppose p:V→R is a seminorm on a real (or complex) vector space
V.
Then
-
1.
p(0)=0,
-
2.
p(v)≥0 for all v∈V.
Proof.
Property 1 follows using homogeneity;
p(0)=p(0⋅0)=|0|p(0)=0. |
Property 2 follows using sublinearity and Property 1;
0=p(0)=p(v-v)≤p(v)+p(-v)=2p(v). |
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Title | basic properties of seminorms |
---|---|
Canonical name | BasicPropertiesOfSeminorms |
Date of creation | 2013-03-22 14:38:57 |
Last modified on | 2013-03-22 14:38:57 |
Owner | matte (1858) |
Last modified by | matte (1858) |
Numerical id | 5 |
Author | matte (1858) |
Entry type | Theorem |
Classification | msc 46B20 |