Linear Algebra Done Right · Problem 6C.16

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · orthogonal-complements

Problem

Suppose

C[-1, 1]

is the vector space of continuous real-valued functions on the interval

[-1, 1]

with inner product given by

\langle f, g \rangle = \int_{-1}^{1} fg

for all

f, g \in C[-1, 1]

. Let

U

be the subspace of

C[-1, 1]

defined by

U = \{f \in C[-1, 1] : f(0) = 0\}.

(a) Show that

U^\perp = \{0\}

. (b) Show that 6.49 and 6.52 do not hold without the finite-dimensional hypothesis.