Hilbert Spaces

Uniform Boundedness Theorem

Let

T_{n}

be a sequence of bounded linear operators

T_{n} : X \to Y

Y

such that for each

x \in X

there exists some

c_{x} \in ℝ^{+}

such that for every

n \in ℕ_{1}

we have

{‖ T_{n} x ‖}_{Y} \leq c_{x}

then there exists some

c \in ℝ^{+}

such that for every

n \in ℕ_{1}

we have that

{‖ T_{n} ‖}_{op} \leq c

Hilbert Adjoint Operator

Let

T : H_{1} \to H_{2}

be a bounded linear operator between hilbert spaces

H_{1}, H_{2}

, then we say that a linear operator

T^{*} : H_{2} \to H_{1}

is a Hilbert adjoint operator if for all

x_{1}, x_{2} \in H_{1}, H_{2}

⟨ T x_{1}, x_{2} ⟩ = ⟨ x_{1}, T^{*} x_{2} ⟩

There Is a Unique Hilbert Adjoint Operator

Given a linear operator

T : H_{1} \to H_{2}

then there exists a unique hilbert adjoint operator

T^{*}

. Moreover it is a bounded linear operator such that

{‖ T^{*} ‖}_{op} = {‖ T ‖}_{op}

TODO: Add the proof here.

Direct Sum

A vector space

X

is said to be the direct sum of two subspaces

Y

and

Z

X

when for every

x \in X

there exists a unique pair of elements

y, z \in Y, Z

such that

x = y + z

and in this case we write

X = Y \oplus︎ Z

Orthogonal in an Inner Product Space

Suppose that

V

is an inner product space, then we say that two vectors

x, y \in V

are orthogonal if

⟨ x, y ⟩ = 0

and we write

x ⟂ y

Orthogonal Projection

For a vector

v

and a closed convex subset of a hilbert space, we denote

v_{U}

to denote this distance minimizing element of

U

called the orthogonal projection of

v

onto

U

Orthogonal Complement of an Inner Product Space

For a subset

U

V

we define the orthogonal complement of

U

to be the set

U = {}

then we use

U^{⊥}

to denote the space of vectors orthogonal of vectors orthogonal to

U

called the orthogonal complement of

u