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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_{d-1}, 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_{d-1}, 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