16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 8.2.16

Inner Product Spaces

Mathematics — Linear Algebra · orthogonal-projection

Problem

Let

V

be an inner product space and

W

a finite-dimensional subspace of

V

. There are (in general) many projections which have

W

as their range. One of these, the orthogonal projection on

W

, has the property that

\|E\alpha\| \leq \|\alpha\|

for every

\alpha

V

. Prove that if

E

is a projection with range

W

, such that

\|E\alpha\| \leq \|\alpha\|

for all

\alpha

V

, then

E

is the orthogonal projection on

W