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Main Question or Discussion Point
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.3 The Tensor Algebra ... ...
I need help in order to get a basic understanding of Definition 10.5 in Section 10.3 ...
Definition 10.5 plus some preliminary definitions reads as follows:
In the above text from Cooperstein, in Definition 10.5, we read the following:
" ... ... An element ##x \in \mathcal{T}(V)## is said to be homogeneous of degree ##d## if ##x \in \mathcal{T}_d (V)## ... ..."
My question is as follows:
How can x be such that ##x \in \mathcal{T}(V)## and ##x \in \mathcal{T}_d (V)## ... does not seem possible to me ... ...
... ... because ... ...
... if ##x \in \mathcal{T}(V)## then ##x## will have the form of an infinite sequence as in the following:
##x = (x_0, x_1, x_2, \ ... \ ... \ , x_{d1}, x_d, x_{d+1}, \ ... \ ... \ ... \ ... ) ##
where ##x_i \in \mathcal{T}_i (V)##
... ... clearly ##x_d## is the ##d##th coordinate of ##x## and so cannot be equal to ##x## ... ..
Can someone please clarify this issue ... clearly I am not understanding this definition ...
Peter
I am focused on Section 10.3 The Tensor Algebra ... ...
I need help in order to get a basic understanding of Definition 10.5 in Section 10.3 ...
Definition 10.5 plus some preliminary definitions reads as follows:
In the above text from Cooperstein, in Definition 10.5, we read the following:
" ... ... An element ##x \in \mathcal{T}(V)## is said to be homogeneous of degree ##d## if ##x \in \mathcal{T}_d (V)## ... ..."
My question is as follows:
How can x be such that ##x \in \mathcal{T}(V)## and ##x \in \mathcal{T}_d (V)## ... does not seem possible to me ... ...
... ... because ... ...
... if ##x \in \mathcal{T}(V)## then ##x## will have the form of an infinite sequence as in the following:
##x = (x_0, x_1, x_2, \ ... \ ... \ , x_{d1}, x_d, x_{d+1}, \ ... \ ... \ ... \ ... ) ##
where ##x_i \in \mathcal{T}_i (V)##
... ... clearly ##x_d## is the ##d##th coordinate of ##x## and so cannot be equal to ##x## ... ..
Can someone please clarify this issue ... clearly I am not understanding this definition ...
Peter
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