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## Main Question or Discussion Point

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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