T denotes the space of infinity sequence of real numbers, .
The function T is linear and isomorphism.
Consider the function
A function is called a linear transformation on if the function 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.
Assume then and .
Substitute the value role="math" localid="1659411768440" as follows.
Now, simplify as follows.
Assume then role="math" localid="1659412647658" .
Substitute the value for as follows.
As , by the definition of linear transformation T is linear.
As the function define from is spanned by means dimension of is 3 and dimension of is 3.
By the definition of polynomial, every polynomial form is described in a unique way means for every there exist a unique such that .
By the definition of isomorphism, the function is isomorphism.
Hence, the transformation localid="1662122176677" is linear and isomorphism.
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