definition of unitary map from vectors space with inner product with induced topology into same vectors space
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of linear map.
- The reader knows a definition of bounded map between normed vectors spaces.
- The reader knows a definition of adjoint map of map from dense subset of vectors space with inner product with induced topology into same vectors space.
Target Context
- The reader will have a definition of unitary map from vectors space with inner product with induced topology into same vectors space.
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:
\( F\): \(\in \{\mathbb{R}, \mathbb{C}\}\), with the canonical field structure
\( V\): \(\in \{\text{ the } F \text{ vectors spaces }\}\), with any inner product, with the topology induced by the metric induced by the norm induced by the inner product
\(*f\): \(: V \to V\), \(\in \{\text{ the bounded linear maps }\}\)
\( f^*\): \(: V^* \to V\), \(= \text{ the adjoint of } f\)
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Conditions:
\(V^* = V \land f^* \circ f = f \circ f^* = id\)
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2: Note
For each \(v, v' \in V\), \(\langle v, v' \rangle = \langle f (v), f (v') \rangle\) is satisfied, because \(\langle f (v), f (v') \rangle = \langle f^* \circ f (v), v' \rangle = \langle id (v), v' \rangle = \langle v, v' \rangle\): for more refer to the proposition that for any map from any vectors space with any inner product with the induced topology into the same vectors space, if the map is unitary, it preserves norm and if the map is any linear surjection that preserves norm, it is unitary.
