I have the following problem:
Consider complex Euclidean spaces H and H′ such that $dim(H) ⩽ dim(H′)$. Let ${|v_i⟩}^{k}_{i=1} ⊂ H$ and ${|w_i⟩}^{k}_{i=1} ⊂ H′$ denote orthonormal sets of vectors.
Show that there exists an isometry $V : H → H′$such that $V |v_i⟩ = |w_i⟩$ for all i ∈ {1,. . . , k}.
I know that if the spaces has equal dimension you could find a unitary operator U such that $U|v_i⟩ = |w_i⟩$ for all i. My question is can we define the the isometry as $$ V=U+\sum |w_j⟩ $$ Where U is a unitary of dimension dim(H) and the sum is of length dim(H')-dim(H)$.
Or maybe something similar?