definition of bounded map between normed vectors spaces
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of normed vectors space.
- The reader knows a definition of map.
Target Context
- The reader will have a definition of bounded map between normed vectors spaces.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( F_1\): \(\in \{\mathbb{R}, \mathbb{C}\}\), with the canonical field structure
\( F_2\): \(\in \{\mathbb{R}, \mathbb{C}\}\), with the canonical field structure
\( V_1\): \(\in \{\text{ the } F_1 \text{ vectors spaces }\}\) with any norm, \(\Vert \bullet \Vert_1\)
\( V_2\): \(\in \{\text{ the } F_2 \text{ vectors spaces }\}\) with any norm, \(\Vert \bullet \Vert_2\)
\(*f\): \(: V_1 \to V_2\)
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Conditions:
\(\exists c \in \mathbb{R} (\forall v_1 \in V_1 (\Vert f (v_1) \Vert_2 \le c \Vert v_1 \Vert_1))\)
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2: Note
\(F_1 \neq F_2\) is allowed.
\(f\) does not need to be linear.
\(f\)'s being linear does not guarantee that \(f\) is bounded: for each unit vector, \(u \in V_1\), \(\Vert f (r u) \Vert_2 = \Vert r f (u) \Vert_2 = \vert r \vert \Vert f (u) \Vert_2 = \vert r \vert \Vert u \Vert_1 / \Vert u \Vert_1 \Vert f (u) \Vert_2 = \Vert r u \Vert_1 \Vert f (u) \Vert_2 \le c_u \Vert r u \Vert_1\) where \(c_u := \Vert f (u) \Vert_2\), but \(c_u\) depends on \(u\) and there is no guarantee that there is a common \(c\) that does not depend on \(u\).
Refer to the proposition that any linear map from any finite-dimensional vectors space with the norm induced by any inner product into any normed vectors space is bounded.
