Adjoint Operator: Proving Unique Adjoint Transformation

[typhoonss821]

I recently teach myself linear algebra with Friedberg's textbook.
And I have a question about adjoint operator, which is on p.367.

Definition Let T : V → W be a linear transformation where V and W are finite-dimensional inner product spaces with inner products <‧,‧> and <‧,‧>' respectively. A funtion T* : W → V is called an adjoint of T if <T(x),y>' = <x,T*(x)> for all x in V and y in W.

Then ,my question is how to prove that there is a unique adjoint T* of T ?

Can anyone give me some tips ? thanks^^

 

[radou]

Assume that there is another adjoint transformation, let's say T**.

 

[Mandark]

To show that there exists such a function, let $$v_1, \ldots , v_n$$ be an orthonormal basis for V, so that $$x = \sum_i \langle x,v_i\rangle v_i$$ for any x in V then we have for all x in V and y in W:

$$\langle T(x), y\rangle ' = \langle T (\sum_i \langle x,v_i\rangle v_i ), y\rangle '$$
$$= \sum_i \langle x, v_i\rangle \langle T(v_i), y\rangle '$$
$$= \langle x, \sum_i \overline{ \langle T(v_i),y\rangle '} v_i\rangle$$
which is in the form that we'd like.

Which shows that $$T^*(y) = \sum_i \overline{\langle T(v_i),y\rangle '} v_i$$ for all y in W works.

 

[typhoonss821]

really appreciate^^