T denotes the space of infinity sequence of real numbers, from P to P.
The function T is linear but not isomorphism.
Consider the function from P to P.
A function D is called a linear transformation on if the function D satisfies the following properties.
An invertible linear transformation is called isomorphism or dimension of domain and co-domain is not same then the function is not isomorphism.
Substitute the value for and for in as follows.
Now, simplify role="math" localid="1659414024917" as follows.
Substitute the value as follows.
As , by the definiti0on of linear transformation T is linear.
Assume , substitute the value in the equation as follows.
As the function T define from P to P, but does not belong to P.
By the definition of isomorphism, the function T is not isomorphism.
Hence, the transformation is linear but not isomorphism.
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