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I know this is a fundamental result in linear algebra, and although it is referenced in my textbook, it does not have a proof for it. I was wondering if someone could help me out:

Let $V$ and $W$ be inner product spaces. If $\dim(V)=\dim(W)$, then $V$ and $W$ are isometrically isomorphic.

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Choose orthonormal bases for the spaces. Map one orthonormal basis bijectively to the other. Extend. – GEdgar Apr 13 '13 at 18:15
up vote 2 down vote accepted

Let $\operatorname{dim}(V)=\operatorname{dim}(W)=n$. Let $\{v_i:i=1,n\}$ be an orthonormal basis of $V$, $\{w_i:i=1,n\}$ be an orthonormal basis of $W$. Following idea suggested by GEdgar define $$ I:V\to W: v\mapsto \sum\limits_{i=1}^n\langle v,v_i\rangle w_i $$ We claim that $I$ the desired isomorphism. Since $\{v_i:i=1,n\}$ is an orthonormal system, then $I(v_i)=w_i$. Since $\{v_i:i=1,n\}$ and $\{w_i:i=1,n\}$ are bases, then the last equality implies that $I$ is a linear isomorphism. It is remains to check it is isometric. Let $v',v''\in V$, then $$ \langle I(v'),I(v'')\rangle =\left\langle\sum\limits_{i=1}^n \langle v',v_i\rangle w_i, \langle v'',v_j\rangle w_j\right\rangle =\sum\limits_{i=1}^n\sum\limits_{j=1}^n\langle v',v_i\rangle \overline{\langle v'',v_j\rangle}\langle w_i,w_j\rangle\\ =\sum\limits_{i=1}^n\sum\limits_{j=1}^n\langle v',v_i\rangle \overline{\langle v'',v_j\rangle} \delta_{ij} =\sum\limits_{i=1}^n\langle v',v_i\rangle \overline{\langle v'',v_i\rangle} =\left\langle v', \sum\limits_{i=1}^n \langle v'',v_i\rangle v_i\right\rangle =\langle v',v''\rangle $$ In particular $$ \Vert I(v)\Vert=\left(\langle I(v),I(v)\rangle\right)^{1/2}=\left(\langle v,v\rangle\right)^{1/2}=\Vert v\Vert $$ so $I$ is isometric. I want to emphsize that this proof may be easily generalized to Hilbert spaces of arbitrary Hilbert dimension.

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