`L^p` spaces

Inequalities

Markov's inequality (T3.1.1, p58)
`f` non-negative measurable function
`0 < t < oo`		 `mu{f >= t} <= (int f dmu)/t`
(C3.1.2, p58)
X an RV on `(Omega, F, P)` `P(|X| >= t) <= (E|X|^r)/(t^r)`
Chebychev's inequality (C3.1.3, p58)
X an RV
`E[x] = mu`
`"Var"(X) = sigma^2 < oo`		 `P(|X - mu| >= k sigma) <= 1/(k^2)`
(C3.1.4, p58)
`phi: RR^+ -> RR^+`, non-decreasing `P(|X| >= t) <= (E phi|X|)/(phi(t))`
Cramer's inequality (C3.1.5, p58)
X an RV, `t>0` `P(|X| >= t) <= inf_(theta >0) (E(e^(theta |X|)))/(e^(theta t))`

A function `phi: (a,b) -> RR` is called convex if `AA 0 <= lambda <= 1, a < x <= y < b` `phi(lambda x + (1-lambda)y <= lambda phi(x) + (1-lambda)phi(y)` (chords rorate counterclockwise). Or equivalently ` (phi(x_2) - phi(x_1))/(x_2-x_1) <= (phi(x_3) - phi(x_2))/(x_3-x_2)` for `a < x_1 < x_2 < x_3 < b`.

If `phi` is twice differentiable on `(a,b)`, and `phi''(x) >= 0` then `phi` is convex.

Holder's inequality T3.1.11, p61
`1 < p < oo, 1/p + 1/q = 1`
`f in L^p(mu), g in L^q(mu)		 `||fg|| <= {: ||f|| :}_p {: ||g|| :}_q`
Cachy-Schwarz inequality C3.1.12, p62
`f,g in L^2(mu)` `||f g|| <= ||f||_2 ||g||_2`
C3.1.13, p62
`1 < p < oo`, `f,g in L^p(mu)` `sum |a_i b_i| c_i <= (sum |a_i|^p c_i)^(1/p) (sum |b_i|^q c_i)^(1/q)`
Minkowski inequality C3.1.14, p62
`1 < p < oo`, `f,g in L^p(mu)` `||f + g||_p <= ||f||_p + ||g||_p`

`L^p` spaces

`L^p` is a vector space. Usual metric is `d_p(f,g) = ||f-g||_p`. If `f` is equivalent to `g` if `f=g "ae" (mu)`. Partitions `L^p(mu)`. Forms a complete metric space.

Dual spaces

`1 <= p < oo`, `1/p + 1/q = 1`, `g in L^q(mu)`, `T_g(f) = int fg du`, `f in L^p(mu)`. Clearly `T_g` is linear, and such a function `L^p(mu) -> RR` is called a *linear functional

The set of all conitunous linear functional on `L^p(mu)` is called the dual space of `L^p(mu)` and denoted `L^p(mu)^**`.

Riesz representation theorem T3.2.3, p57
`1 <= < oo`
`1/q + 1/p = 1`
`T: L^p(mu) -> RR` linear and continuous		 `EE g in L^q(mu) "st" T(f) = T_g(f)`.
Not valid for `p=oo`.
`(L^p(mu)^**, ||T||)` is a normed linear vector space, and more `||phi(T)||_q = ||T|| AA T in (L^p(mu)^**`, it is an isometry.

Banach and Hilbert spaces

Banach spces

A Banach space is a complete normed vector space. All `L^p(mu)` spaces are Banach spaces. A closed subspace of a Banach space is also a Banach space.

A norm must satisfy:

A linear transformation is a function `f: V_1 -> V_2` if `a_1, a_2 in RR, x,y in V_1 => T(alpha_1 x + alpha_2 y) = alpha_1 T(x) + alpha_2 T(y)`.

Hilbert spaces

A vector space is a real innerproduct space if `EE f: V xx V -> RR`, denoted by `f(x,y) = (:x,y:)`, that satisfies:

A Hilbert space is a complete real inner product space. Every Hilbert space is a `L^2(Omega, F, mu)` space for some `(Omega, F, mu)`. Called separable if `EE` dense countable subset.

Orthogonal vectors `x _|_ y iff (: x, y :) = 0`. Orthnormal if `||x|| = 1`. If `B sub H` is orthogonal and `H` is separable, then `B` is at most countable. Can convert any set to an orthonormal set using Gram-Schmidt organalisation.

The fourier coefficients of a vector `x in V` wrt ON `B sub V` is `{(:x, b:): b in B}`.

Bessel’s inequality: `{b_i}_{i>=1}` ON in an IPS `(V, (: *, * :))`, `AA x in V sum (: x, b_i :)^2 <= ||x||^2`.

Following are equivalent:

Riesz representation theorem: `H` a separable Hilbert space, then every bounded linear functional `T: H -> RR` can be represented as `T ~= T_(x_0) = (: y, x_0 :)` for some `x_0 in V`.