FA

Complex VS

NVS

ex) $(\mathbb{C}^n,\lVert \cdot\rVert_p)$, $(l^p(\mathbb{N}),\lVert\cdot\rVert_p)$, $(L^p(\mathbb{R}),\lVert\cdot\rVert)$

(Singular Value Decomposition Thm.) $A=UDV^*$

Functional Calculus - $f(|A|):=Vf(D)V^*$ ($f\in C[0,\infty)$)

$(M_n(\mathbb{C}),\lVert\cdot\rVert_p)$(Schatten p-norm) NVS

$\lVert\cdot\rVert_\infty =\lVert\cdot \rVert_{op}$

Measure Theory

Measure space

ex) $(\mathbb{R}^n,\mathfrak{M},m)$, $(X,2^X,|\cdot|)$, $(X,2^X,\delta_x)$

non mea’ble ex) Vitali set - choose one elt. in [0,1] for each in $\mathbb{R}/\mathbb{Q}$

mea’ble but non Borel ex) for Cantor set C — 3-adic expression only consists of 0 and 2 — set non mea’ble $F\subset g(C)$, consider $g^{-1}(F)$

Lebesgue Integral, $L^p$

Banach space

ex) NVS ex above + $(C(K,Y),\lVert\cdot\rVert_\infty)$($K\subset \mathbb{R}^n$ compact, $Y$ Banach), $(C_0(X),\lVert\cdot\rVert)$($X$ lcpt)

Hilbert Space

separable Hilbert space -dimension countable

Hilbert ex and only) $L^2(X)$

Fourier Series

Fourier analysis on $\mathbb{T}$

Parseval Identity

$B(X,Y)$

$X$ metrizable ⇒ $T$ conti↔ $T$ bdd.

non-ex) $T:e_n\mapsto ne_n$