Problem 3. An isometry between inner-product spaces V and W is alinear
operator L in B (V ,W) that preserves norms and inner-products. Ifx, y in V
and if L is an isometry, then we have _W = _V .
Suppose that V and W are both real, n-dimensional inner-productspaces.
Thus the scalar field for both is R and both of them have a basisconsisting of
n elements. Show that V and W are isometric by demonstrating anisometry
between them.
Hint: take both bases, and cite some linear algebra result thatsays that
you can orthonormalize them. Prove (or cite someone to convince me)that you
can define a linear function by specifying its action on a basis.Finally, define
your isometry by deciding what it should do on an orthonormal basisfor V , and
prove that it preserves inner-products (and thus norms).
