Page 240 - Advanced Linear Algebra
P. 240
224 Advanced Linear Algebra
) )
))#~#
V
for all #= .
13. Prove that an orthonormal set is a Hilbert basis for a finite-dimensional
E
vector space if and only if Parseval's identity holds for all Á # $ = , that
=
is, if and only if
º#Á $» ~ ´#µ h ´$µ V E V E
for all #Á $ = .
14. Let " ~ ² ÁÃÁ ³ and # ~ ² Á ÃÁ ³ be in s . The Cauchy–Schwarz
inequality states that
( ( bÄb ² bÄb ³² b Äb ³
Prove that we can do better:
(
(
² b Äb ³ ² bÄb ³² b Äb ³
(
(
15. Let be a finite-dimensional inner product space. Prove that for any subset
=
? = of , we have ? ~ ² span ? ³ .
16. Let F 3 be the inner product space of all polynomials of degree at most 3,
under the inner product
B
º ²%³Á ²%³» ~ ²%³ ²%³ c% %
cB
Apply the Gram–Schmidt process to the basis ¸ Á %Á %Á %¹ , thereby
computing the first four Hermite polynomials at least up to a
(
)
multiplicative constant .
17. Verify uniqueness in the Riesz representation theorem.
18. Let be a complex inner product space and let be a subspace of .
=
:
=
Suppose that # = is a vector for which º#Á » b º Á #» º Á » for all
:. Prove that # : .
19. If = and > are inner product spaces, consider the function on = ^ >
defined by
º²# Á $ ³Á ²# Á $ ³» ~ º# Á # » b º$ Á $ »
Is this an inner product on = > ^ ?
(
20. A normed vector space over or is a vector space over or d )
s
d
s
together with a function ))¢= ¦ s for which for all "Á# = and scalars
we have
a ) )) ~ ( () )
#
#
b ) ) )"b# ) ) )" b # )
c ))#~ if and only if # ~
)
(
If = is a real normed space over s ) and if the norm satisfies the
parallelogram law
