Book Recommendation for Linear Algebra using analysis. Today in my Analysis class my teacher proved that (real)symmetric bilinear forms (in an inner-product space) are orthogonally diagonalizable using compactness, differentiablity of innerproducts and other concepts of analysis and then he proceeded to prove that the k'th largest eigenvalue can be derived by 
$$\min_{W\subset V,\dim W=k} \max_{x\in W , \|x\|=1} \langle Ax,x\rangle$$
where the bilinear form is $B(x,y)=\langle Ax,y\rangle$.
Can anyone suggest me some books which have these kind of "analysis" flavoured linear algebra? 

You can try Hubbard and Hubbard's "Vector Calculus, Linear Algebra, and Differential Forms."
