Understanding Bra Ket Correspondence and Proving (1.8) Transformation

[Kashmir]

I can't follow how the above argument leads to (1.8).

I am able to prove it only if I can show $\langle a \mid c\rangle\langle b+c\rangle=(\langle a|+\langle b|) c\rangle$ but I don't understand why the bra transformations <P| ,<Q| obey
(<P|+ <Q|)x = <P|x + <Q|x .
Is it an assumption?

Please help me

 

[PeroK]

View attachment 290057

I can't follow how the above argument leads to (1.8).

I am able to prove it only if I can show $\langle a \mid c\rangle+\langle b+c\rangle=(\langle a|+\langle b|) c\rangle$ but I don't understand why the bra transformations <P| ,<Q| obey
(<P|+ <Q|)x = <P|x + <Q|x .
Is it an assumption?

Please help me

It's not an assumption. In order to make progress with QM - and especially Dirac notation - you are going to have to learn some formal linear algebra and, in particularm how to use defined properties to construct proofs.

(1.8) follows directly from the conjugate linearity of the inner product.

 

[Kashmir]

It's not an assumption. In order to make progress with QM - and especially Dirac notation - you are going to have to learn some formal linear algebra and, in particularm how to use defined properties to construct proofs.

(1.8) follows directly from the conjugate linearity of the inner product.

Perhaps you mean this:

$(a+b, c)=(|a+b\rangle,|c\rangle)=\langle a+b \mid c\rangle -(1)$Also
$(a+b, c)=(a, c)+(b, c)=\langle a \mid c\rangle+\langle b \mid c\rangle-(2)$

From equations 1,2 we have

$\langle a+b \mid c\rangle=\langle a \mid c\rangle+\langle b \mid c\rangle$ and not $\langle a+b \mid c\rangle=(\langle a|+\langle b|) c\rangle$ which is my doubt.

 

[PeroK]

Perhaps you mean this:

$(a+b, c)=(|a+b\rangle,|c\rangle)=\langle a+b \mid c\rangle -(1)$Also
$(a+b, c)=(a, c)+(b, c)=\langle a \mid c\rangle+\langle b \mid c\rangle-(2)$

From equations 1,2 we have

$\langle a+b \mid c\rangle=\langle a \mid c\rangle+\langle b \mid c\rangle$ and not $\langle a+b \mid c\rangle=(\langle a|+\langle b|) c\rangle$ which is my doubt.

These constructions are invalid: you can't mix orthodox linear algebra notation with bras and kets. It's one or the other. Note how careful the author of the section you posted was to keep the terminology consistent.

In any case, I thought the question was how to prove that:$$\langle u|\alpha^* \ \leftrightarrow \ \alpha |u\rangle$$What that means is: if $\langle u|$ is the bra corresponding to the ket $|u \rangle$, then the bra $\langle u|\alpha^*$corresponds to the ket $\alpha |u\rangle$.

Note that $\alpha$ is a scalar - you also seem to be confused by the roles of scalars and kets. You may need a course in linear algebra before you can proceed any further with QM.