Adjoint space

I have a question which relate adjoint space. That is: ” Is C^*[0,1] linear isomorphism with C[0,1]?” C[0,1] is space of continous function in [0,1]. If we choose inner product of C[0,1] is <,>: C[0,1] \longrightarrow\Bbb{R}, by <f,g>=\int_0^1  f(x)g(x)dx then C^*[0,1] = C[0,1] (linear isomorphism)?

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