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I have an issue/problem that relates to Bland initial treatment of external direct sums including Proposition 2.1.5 ... especially Bland's definition of the sum of a family of mappings ...
Bland's text on this is as follows:
In the above text by Bland we read the following:
" ... ... We now need the concept of a family of mappings. If ##f_\alpha \ : \ M_\alpha \rightarrow N## is an Rlinear mapping for each ##\alpha \in \Delta##, where ##N## is a fixed Rmodule, then ##f \ : \ \bigoplus_\Delta M_\alpha \rightarrow N## defined by ##f( ( x_\alpha ) ) = \sum_\Delta f_\alpha (x)## ... ... "
But in the last sentence of the proof of Proposition 2.1.5 ( ... again, see above text by Bland ... ) we read:
" ... ... If ##( x_\alpha ) \in \bigoplus_\Delta M_\alpha##, then ##f( ( x_\alpha ) ) = \sum_\Delta f_\alpha (x_\alpha)## ... ... "
So ... in the text above the Proposition we have ... ##f( ( x_\alpha ) ) = \sum_\Delta f_\alpha (x)## ... ... and in the proof of the proposition we have ##f( ( x_\alpha ) ) = \sum_\Delta f_\alpha (x_\alpha)## ...
... which of these is correct ... or in some strange way, are they both correct ...
... I note that x is mentioned in the definition of the canonical injections above ..
Can someone please clarify ... ?
But ... if ##f## is defined by ##f( ( x_\alpha ) ) = \sum_\Delta f_\alpha (x_\alpha)## ... then I have a further problem ...
... we know that ##f \ : \ M_\alpha \rightarrow N## ... that is the domain of ##f_\alpha## is ##M_\alpha## ... BUT ...PROBLEM ... ##( x_\alpha ) \in \bigoplus_\Delta M_\alpha## and ##( x_\alpha ) \notin M_\alpha## ...
... can someone please clarify ?
Hope someone can help ...
Peter
Bland's text on this is as follows:
In the above text by Bland we read the following:
" ... ... We now need the concept of a family of mappings. If ##f_\alpha \ : \ M_\alpha \rightarrow N## is an Rlinear mapping for each ##\alpha \in \Delta##, where ##N## is a fixed Rmodule, then ##f \ : \ \bigoplus_\Delta M_\alpha \rightarrow N## defined by ##f( ( x_\alpha ) ) = \sum_\Delta f_\alpha (x)## ... ... "
But in the last sentence of the proof of Proposition 2.1.5 ( ... again, see above text by Bland ... ) we read:
" ... ... If ##( x_\alpha ) \in \bigoplus_\Delta M_\alpha##, then ##f( ( x_\alpha ) ) = \sum_\Delta f_\alpha (x_\alpha)## ... ... "
So ... in the text above the Proposition we have ... ##f( ( x_\alpha ) ) = \sum_\Delta f_\alpha (x)## ... ... and in the proof of the proposition we have ##f( ( x_\alpha ) ) = \sum_\Delta f_\alpha (x_\alpha)## ...
... which of these is correct ... or in some strange way, are they both correct ...
... I note that x is mentioned in the definition of the canonical injections above ..
Can someone please clarify ... ?
But ... if ##f## is defined by ##f( ( x_\alpha ) ) = \sum_\Delta f_\alpha (x_\alpha)## ... then I have a further problem ...
... we know that ##f \ : \ M_\alpha \rightarrow N## ... that is the domain of ##f_\alpha## is ##M_\alpha## ... BUT ...PROBLEM ... ##( x_\alpha ) \in \bigoplus_\Delta M_\alpha## and ##( x_\alpha ) \notin M_\alpha## ...
... can someone please clarify ?
Hope someone can help ...
Peter
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