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My question is as it says in the title really. I've been reading Nakahara's book on geometry and topology in physics and I'm slightly stuck on a part concerning adjoint mappings between vector spaces. It is as follows:

Let [itex]W=W(n,\mathbb{R})[/itex] be a vector space with a basis [itex]\lbrace\mathbf{f}_{\alpha}\rbrace[/itex] and a vector space isomorphism [itex]G:W\rightarrow W^{\ast}[/itex].

Given a map [itex]f:V\rightarrow W[/itex] we may define the [itex]\bf{adjoint}[/itex] of [itex]f[/itex], denoted by [itex]\tilde{f}[/itex], by [tex]G(\mathbf{w},f\mathbf{v}) =g(\mathbf{v},\tilde{f}\mathbf{w})[/tex] where [itex]\mathbf{v}\in V[/itex] and [itex]\mathbf{w}\in W[/itex], [itex]g(\cdot,\cdot)[/itex] is the inner product between the two vectors [itex]\mathbf{v}[/itex] and [itex]\tilde{f}\mathbf{w}[/itex].

He then goes on to say that "it is easy to see from this, that [itex]\widetilde{(\tilde{f})}=f[/itex]".

I'm having trouble showing that this is true given the definitions above.

Let [itex]W=W(n,\mathbb{R})[/itex] be a vector space with a basis [itex]\lbrace\mathbf{f}_{\alpha}\rbrace[/itex] and a vector space isomorphism [itex]G:W\rightarrow W^{\ast}[/itex].

Given a map [itex]f:V\rightarrow W[/itex] we may define the [itex]\bf{adjoint}[/itex] of [itex]f[/itex], denoted by [itex]\tilde{f}[/itex], by [tex]G(\mathbf{w},f\mathbf{v}) =g(\mathbf{v},\tilde{f}\mathbf{w})[/tex] where [itex]\mathbf{v}\in V[/itex] and [itex]\mathbf{w}\in W[/itex], [itex]g(\cdot,\cdot)[/itex] is the inner product between the two vectors [itex]\mathbf{v}[/itex] and [itex]\tilde{f}\mathbf{w}[/itex].

He then goes on to say that "it is easy to see from this, that [itex]\widetilde{(\tilde{f})}=f[/itex]".

I'm having trouble showing that this is true given the definitions above.

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