Answer
False
Work Step by Step
Let vector space $C^0[a,b]$ has $a,b \in R$ and $f,g \in C^0[a,b]$
One inner product of this vector space is $=\int ^b_a f(x)g(x)dx$
We can define as $=\int ^d_c f(x)g(x)dx$
Hence, we can see the mapping above is also an inner product of $C^0[a,b]$
Thus, the statement is false.