4.3 KiB
Inner product spaces
Definition 1: a vector space
X
over a fieldF
is an inner product space if an inner product\langle \cdot, \cdot \rangle: X \times X \to F
is defined onX
satisfying
\forall x \in X: \langle x, x \rangle \geq 0
,\langle x, x \rangle = 0 \iff x = 0
,\forall x, y \in X: \langle x, y \rangle = \overline{\langle y, x \rangle}
,\forall x, y \in X, \alpha \in F: \langle \alpha x, y \rangle = \alpha \langle x, y \rangle
,\forall x, y, z \in X: \langle x + y, z \rangle = \langle x, z \rangle + \langle y, z \rangle
.
Similar to the case in normed spaces we have the following proposition.
Proposition 1: an inner product
\langle \cdot, \cdot \rangle
on a vector spaceX
defines a norm\|\cdot\|
onX
given by
|x| = \sqrt{\langle x, x \rangle},
for all
x \in X
and is called the norm induced by the inner product.
??? note "Proof:"
Will be added later.
Which makes an inner product space also a normed space as well as a metric space, referring to proposition 1 in normed spaces.
Definition 2: a Hilbert space
H
is a complete inner product space with its metric induced by the inner product.
Definition 2 makes a Hilbert space also a Banach space, using proposition 1.
Properties of inner product spaces
Proposition 2: let
(X, \langle \cdot, \cdot \rangle)
be an inner product space, then
| x + y |^2 + | x - y |^2 = 2\big(|x|^2 + |y|^2\big),
for all
x, y \in X
.
??? note "Proof:"
Will be added later.
Proposition 2 is also called the parallelogram identity.
Lemma 1: let
(X, \langle \cdot, \cdot \rangle)
be an inner product space, then
\forall x, y \in X: |\langle x, y \rangle| \leq \|x\| \cdot \|y\|
,\forall x, y \in X: \|x + y\| \leq \|x\| + \|y\|
.
??? note "Proof:"
Will be added later.
Statement 1 in lemma 1 is known as the Schwarz inequality and statement 2 is known as the triangle inequality and will be used throughout the section of inner product spaces.
Lemma 2: let
(X, \langle \cdot, \cdot \rangle)
be an inner product space and let(x_n)_{n \in \mathbb{N}}
and(y_n)_{n \in \mathbb{N}}
be sequences inX
, if we havex_n \to x
andy_n \to y
asn \to \infty
, then
\lim_{n \to \infty} \langle x_n, y_n \rangle = \langle x, y \rangle.
??? note "Proof:"
Will be added later.
Completion
Definition 3: an isomorphism
T
of an inner product space(X, \langle \cdot, \cdot \rangle)_X
onto an inner product space(\tilde X, \langle \cdot, \cdot \rangle)_{\tilde X}
over the same fieldF
is a bijective linear operatorT: X \to \tilde X
which preserves the inner product
\langle Tx, Ty \rangle_{\tilde X} = \langle x, y \rangle_X,
for all
x, y \in X
.
As a first application of lemma 2, let us prove the following.
Theorem 1: for every inner product space
(X, \langle \cdot, \cdot \rangle)_X
there exists a Hilbert space(\tilde X, \langle \cdot, \cdot \rangle)_{\tilde X}
that contains a subspaceW
that satisfies the following conditions
W
is an inner product space isomorphic withX
.W
is dense inX
.
??? note "Proof:"
Will be added later.
Somewhat trivially, we have that a subspace M
of an inner product space X
is defined to be a vector subspace of X
taken with the inner product on X
restricted to M \times M
.
Proposition 3: let
Y
be a subspace of a Hilbert spaceX
, then
Y
is complete\iff
Y
is closed inX
,- if
Y
is finite-dimensional, thenY
is complete,Y
is separable ifX
is separable.
??? note "Proof:"
Will be added later.
Orthogonality
Definition 4: let
(X, \langle \cdot, \cdot \rangle)
be an inner product space, a vectorx \in X
is orthogonal to a vectory \in X
if
\langle x, y \rangle = 0,
and we write
x \perp y
.
Furthermore, we can also say that x
and y
are orthogonal.
Definition 5: let
(X, \langle \cdot, \cdot \rangle)
be an inner product space and letA, B \subset X
be subspaces ofX
. ThenA
is orthogonal toB
if for everyx \in A
andy \in B
we have
\langle x, y \rangle = 0,
and we write
A \perp B
.
Similarly, we may state that A
and B
are orthogonal.