Functional analysis is the study of maps on vector spaces, aka operators, especially linear operators and, if the space is endowed with a topology, continuous operators. Functional analysis was initially the analysis of functionals, i.e. scalar-valued operators. Vector spaces of particular interest are spaces of functions with certain properties, e.g. ones that are continuous, measurable, differentiable, linear, or satisfying a dynamical system. Vector spaces in functional analysis are often endowed with more structures, e.g. topology, norm, inner product, or measure. The most highly developed parts of functional analysis are the theories of normed space, Banach space, and linear operators on such spaces.

Notes on Normed Linear & Banach Space

## Topological Vector Space

Topological vector space $(V, (+, \cdot_\mathbb{F}), \mathcal{T})$ is a vector space over a topological field $(\mathbb{F}, \mathcal{T}_F)$ equipped with a topology such that vector addition and scalar multiplication are continuous maps.

Total subset or fundamental subset in a topological vector space is a subset whose span is a dense subspace: $\overline{\text{Span}(A)} = V$.

Compact operator between topological vector spaces is an operator that maps bounded subsets to totally-bounded/pre-compact subsets.

Closed operator between topological vector spaces is an operator whose graph is a closed subset of the product space: $\Gamma(T) \in \mathcal{T}^∗_{V \times W}$; or equivalently, every convergent sequence with a convergent image converges to a point on the graph, $\lim v_n = v$, $\lim T v_n = w$, then $T v = w$. Note that closed operator (between topological vector spaces) and close map (between topological spaces) are different.

Continuous operator $T: M \to W$ between (subsets of) topological vector spaces is a continuous map: $T \in C(M, W)$. Weakly continuous operator... Strongly continuous operator... Continuous functional $f: V \to \mathbb{F}$ on a topological vector space is a continuous operator to its underlying scalar field: $f \in C(V)$.

### Dual Space and Dual Operator

Dual space or adjoint space $V^\#$ of a topological vector space is the vector space of continuous linear functionals: $V^\# = \mathcal{L}(V) \cap C(V)$. Similar to vector space, we will call any element in the dual space of a topological vector space a covector, i.e. a continuous linear functional.

Locally convex topology on a vector space is one that has a local base at the origin consisting of balanced, convex, and absorbing sets. Locally convex space is a topological vector space whose topology is Hausdorff locally convex. The dual space of a locally convex space separate the points of the space.(?) Semi-norm $p: V \to \mathbb{R}_{\ge 0}$ is a mapping that meets the requirements of a norm except for non-degeneracy: (1) absolute homogeneity: $p(a v) = |a| p(v)$; (2) triangle axiom: $p(v + w) \le p(v) + p(w)$. Generalized Hahn-Banach Theorem [@Bohnenblust and Sobczyk, 1938]: Every linear functional on a subspace of a vector space that is bounded from above by a semi-norm on the space has a linear extension to the space that remains bounded from above by the semi-norm: $\omega \in S^∗$, $|\omega| \le p$, then $\exists \tilde\omega \in V^∗$, $|\tilde\omega| \le p$. Weak topology $\sigma(V, F)$ on a topological vector space given a subset $F$ of its adjoint space $V^\#$ is the locally convex topology generated by the family of semi-norms $\{|f(\cdot)|\}_{f \in F}$; it is a Hausdorff topology if and only if the subset is a total subset, i.e. the subset separates the points of the space. Weak∗ topology on the adjoint space of a topological vector space is the weakest topology for which all the evaluation mappings $T_v: V^\# \to \mathbb{F}$, $T_v f = f(v)$, $v \in V$ are continuous.

Scalar product $q: V \times W \to \mathbb{F}$ on the Cartesian product of two vector spaces over the same field is a nondegenerate bilinear scalar-valued function: $q(a v + b v', w) = a q(v, w) + b q(v', w)$, $q(v, \cdot) = 0 \implies v = 0$, and both apply analogously for the second variable. Dual pair of vector spaces $(V, W, q(\cdot, \cdot))$ is a pair of vector spaces $(V, W)$ over the same field $\mathbb{F}$, together with a scalar product $q(\cdot, \cdot)$. Weak topology defined by a dual pair of vector spaces over the same topological field is the weakest topology on one space such that the scalar product with any vector in the other space is a continuous linear functional: $\{q(\cdot, w) : w \in W\} \subset V^\#$. Strong topology defined by a dual pair of vector spaces over the same topological field is the topology on one space of uniform convergence on the bounded subsets of the other space for the weak topology defined by the dual pair.

Strong dual space $(V^\#, \mathcal{T})$ of a topological vector space $V$ is its dual space $V^\#$ with the strong topology. Second dual space or bi-dual space $V^{\#\#}$ of a Hausdorff locally convex space $V$ is the dual space of its strong dual space. Barrelled space is a locally convex space with several properties of Banach spaces and Fréchet spaces. Semi-reflexive space is a Hausdorff locally convex space that coincides with its second dual: $V = V^{\#\#}$.

Dual operator, adjoint operator, or conjugate operator $T^\#: W^\# \to V^\#$ of a linear operator $T: V \to W$ between locally convex spaces is a linear operator between the strong dual spaces $W^\#$ and $V^\#$ such that $(w^\#, v^\#) \in T^\#$ if and only if $\exists (v, w) \in T$: $(w, w^\#) = (v, v^\#)$. The adjoint operator of a continuous linear operator is also continuous.

## Normed Space

Norm $\|\cdot\|: V \to \mathbb{R}_{\ge 0}$ on a real or complex vector space $V$ is a nonnegative operator that satisfies: (1) non-degeneracy: nonzero vector has nonzero norm, $\|v\| = 0 \implies v = 0$; (2) absolute homogeneity (of degree 1): scaled vector has absolutely scaled norm, $\|a v\| = |a| \|v\|$; (3) subadditivity, or triangle inequality: norm of sum does not exceed sum of norms, $\|v + w\| \le \|v\| + \|w\|$. Strictly convex norm is a norm such that distinct unit vectors sum to vectors of length less than two: $v, w \in B_1(0)$, $v \ne w$, then $\|v + w\| < 2$. Strongly equivalent norms on a vector space are norms whose ratio has positive bounds: $\forall v \in V$, $a \le \frac{\|v\|_\alpha}{\|v\|_\beta} \le b$. All norms on a finite-dimensional vector space are equivalent. Equivalent norms on a vector space induce the same topology. Strong equivalence of norms preserves differentiability.

Normed space $(V, (+, \cdot_\mathbb{F}), \|\cdot\|)$ is a vector space endowed with a norm. Strictly convex normed space is a vector space endowed with a strictly convex norm. An infinite-dimensional vector space can be endowed with a norm that is undefined on a subspace; in such cases, we regard the normed space to be the intersection of the vector space and the domain of the norm. For example, the real function space endowed with the uniform norm, $(C(\mathbb{R}), \|\cdot\|_\infty)$, is understood to only contain the bounded real functions. Norm specifies the length of each element of a vector space. A norm induces a metric on the vector space, $d(v, w) = \|v − w\|$, which in turn induces a topology on the vector space; as a result, every normed space is a topological vector space. The norm of a normed space is not a linear functional, but it is a continuous functional because the induced metric is a continuous function. A subset in a finite-dimensional normed space is a compact topological space if and only if it is a closed subset and a bounded metric subspace. A normed space is finite-dimensional if its closed unit ball is compact. Reflexive normed space is a normed space whose canonical embedding is surjective: $\xi(V) = V^{\#\#}$. A separable normed space with a nonseparable dual space cannot be reflexive. A normed space is separable if its dual space is separable.

Banach space is a normed space that is complete in the metric induced by the norm. (The term is named in recognition of Stefan Banach, who wrote the first monograph on the subject [@Banach1932].) Subspace $(S, (+, \cdot_\mathbb{F}), \|\cdot\|)$ of a normed space is a normed space consisting of a linear subspace and the norm restricted to the subspace. A subspace of a Banach space is a Banach space if and only if it is a closed subset. Every finite-dimensional subspace of a normed space is a Banach space, and thus a closed subset; in particular, every finite-dimensional normed space is a Banach space. Completion of normed space: Every normed space is isometric to a dense subspace of a unique Banach space up to isometries. Completion $\widehat V$ of a normed space is the Banach space with a dense subspace isometric to the normed space.

Linear isometry or normed space isomorphism between normed spaces is a vector space isomorphism that preserves the norms: $\forall v \in V$, $\|T v\| = \|v\|$.

### Convergence

A sequence in a normed space converges to a vector in the space if and only if the length of the difference vector converges to zero: $\lim_{n \to \infty} \|v_n - v\| = 0$. Uniform convergence of a sequence of real functions on an interval is convergence in the uniform-normed space of real function on the interval.

A sequence in a normed space weakly converges to a vector in the space if every covector maps the sequence to a scalar sequence that converges to its value at the vector: $\forall \omega \in V^\#$, $\lim_{n \to \infty} \omega v_n = \omega v$. We say the sequence is a weakly convergent sequence in the space, and the vector is the weak limit of the sequence: $v_n \overset{w}{\to} v$. In this context, a convergent sequence in a normed space is called a strongly convergent sequence.

### Series

Convergent series in a topological vector space is a convergent sequence of partial sums of a sequence in the space. Sum $\sum_{i=1}^\infty x_i$ of a convergent series in a topological vector space is the limit of the series: $\sum_{i=1}^\infty x_i = \lim_{n \to \infty} \sum_{i=1}^n x_i$. The notation for the sum of a convergent series is commonly used to denote any series, which is understood to be nonexistent if the series does not converge, and synonymous to the series if it converges.

Absolutely convergent series in a normed space is a series such that the corresponding series of norms converges to a positive number: $\sum_{i=1}^\infty \|x_i\| < \infty$. All absolutely convergent series in a normed space are convergent if and only if the space is Banach. Conditionally convergent series of real numbers is a convergent series that is not absolutely convergent. Convergence of a series depends on the order of the underlying sequence. Riemann Series Theorem or Riemann rearrangement theorem: If a real sequence determines a conditionally convergent series, then it can be rearranged to have a series that diverges or converges to any real number; in particular, its positive subsequence and negative subsequence both diverge.

Schauder basis $(e_i)_{i \in \mathbb{N}}$ for a normed space is a linearly independent sequence such that every vector in the space is the sum of a unique series of scaled vectors in the sequence: $\forall v \in V$, $\exists! (a_i)$: $\sum_{i \in \mathbb{N}} a_i e_i = v$. A normed space is separable if it has a Schauder basis; almost all known separable Banach spaces have a Schauder basis, but [@Enflo1973] constructed a separable Banach space with no Schauder basis. Expansion $\sum_{i \in \mathbb{N}} a_i e_i$ of a vector in a normed space w.r.t. a Schauder basis is the series of scaled basis vectors that converges to the vector.

## $l^p$ Space

$l^p$ and $l^\infty$ spaces can be seen as the sequence version of $L^p$ and $L^\infty$ spaces.

l^p norm $\|\cdot\|_p$ of a scalar sequence, $p \in [1, \infty)$, is the principal p-th root of the sum of the p-th power of the absolute values, if it exists: $\| (x_i)_{i \in \mathbb{N}} \|_p = \sqrt[q]{\sum_{i=1}^\infty |x_i|^p}$. For example, $l^1$ norm is the sum of absolute values: $\|x\|_1 = \sum_{i=1}^\infty |x_i|$; and $l^2$ norm is the square root of the sum of squares: $\|x\|_2 = \sum_{i=1}^\infty |x_i|^2$. l^p metric $d_p(\cdot, \cdot)$ is the metric induced from an $l^p$ norm. The $l^2$ norm can be extended to an inner product of sequences of scalars: $\langle (x_i), (y_i) \rangle = \sum_{i=1}^\infty x_i \bar y_i$. l^p space $(l^p, \|\cdot\|_p)$ is an $l^p$-normed space of scalar sequences: $l^p = \{(x_i)_{i \in \mathbb{N}} : \sum_{i=1}^\infty |x_i|^p < \infty\}$. The $l^2$ space is also called the Hilbert sequence space [@Hilbert1912], which is a Hilbert space.

Conjugate indices $(p, q)$ is a pair of positive numbers whose reciprocals sum to one: $1/p + 1/q = 1$. Holder inequality for sums: The inner product of two sequences is no greater than the product of the $l^p$ norm of one and the $l^q$ norm of the other, for any conjugate indices $(p, q)$: $\sum_{i=1}^\infty |x_i y_i| \le \left(\sum_{i=1}^\infty |x_i|^p\right)^{1/p} \left(\sum_{i=1}^\infty |y_i|^q\right)^{1/q}$. Cauchy-Schwarz inequality for sums is the Holder inequality for sums where $p = q = 2$: the inner product of two sequences is no greater than the product of their $l^2$ norms. Minkowski inequality for sums is the triangle inequality of $l^p$ norms: $\left(\sum_{i=1}^\infty |x_i + y_i|^p\right)^{1/p} \le \left(\sum_{i=1}^\infty |x_i|^p\right)^{1/p} + \left(\sum_{i=1}^\infty |y_i|^p\right)^{1/p}$.

l^∞ norm $\|\cdot\|_\infty$ of a scalar sequence is the supremum of its absolute values: $\| (x_i)_{i \in \mathbb{N}} \|_\infty = \sup_{i \in \mathbb{N}} |x_i|$. l^∞ space $(l^\infty, \|\cdot\|_\infty)$ is an $l^\infty$-normed space of scalar sequences: $l^\infty = \{(x_i)_{i \in \mathbb{N}} : \sup_{i \in \mathbb{N}} |x_i| < \infty\}$.

The dual space of $l^1$ is $l^\infty$: $(l^1)^\# = l^\infty$. The dual space of $l^p$, $p \in (1, \infty)$, is $l^q$ such that $(p, q)$ are conjugate indices: $1/p + 1/q = 1$ then $(l^p)^\# = l^q$.

The $l^1$ space is separable; the $l^\infty$ space is not separable.

## Linear Operator

Operator norm of an operator between normed spaces is the supremum of its norm scaling ratio, or equivalently, the supremum of norms of the image of the unit ball: $\|T\| = \sup_{\|v\| \le 1} \|T v\|$, i.e. $\|T\| = \sup_{v \in V} \frac{\|T v\|}{\|v\|}$. Bounded operator between normed spaces is an operator with a finite operator norm: $\|T\| < \infty$. The norm of a normed space is a bounded functional: $\|\|\cdot\|\| = 1$. Integral operators are often bounded if the kernel satisfies certain boundedness condition, but the differentiation operator is typically unbounded. Space of bounded linear operators $B(V, W)$ between normed spaces is the vector space of linear operators endowed with the operator norm: $B(V, W) = (\mathcal{L}(V, W), \|\cdot\|)$. A linear operator between normed spaces is continuous if it is continuous at one point. A linear operator between normed spaces is continuous if and only if it is bounded: if $T \in \mathcal{L}(V, W)$, then $T \in C(V, W)$ iff $T \in B(V, W)$; in particular, the dual space of a normed space equals the bounded linear functional space: $V^\# = B(V)$. A bounded linear operator space is Banach if the codomain is Banach: $\widehat W = W$ then $\widehat B(V, W) = B(V, W)$; in particular, the dual space of a normed space is Banach: $\widehat B(V) = B(V)$, i.e. $\widehat{V^\#} = V^\#$. Every linear operator on a finite-dimensional normed space is bounded: $\dim V < \infty$ then $\mathcal{L}(V, W) = B(V, W)$ and hence $\mathcal{L}(V, W) = C(V, W)$.

Completely-continuous operator from a Banach space to a normed space is a bounded linear operator that maps weakly-convergent sequences to convergent sequences. Every compact operator is completely-continuous; a completely-continuous operators is compact if its codomain is also Banach. The adjoint operator of a completely-continuous operator is also completely-continuous.

The four fundamental theorems of linear operators on normed or banach spaces are Hahn-Banach theorem, uniform boundedness theorem, open mapping theorem, and closed graph theorem.

### Hahn-Banach Theorem

Bounded linear extension: Every bounded linear operator from a subspace of a normed space to a Banach space has a unique extension to the closure of its domain, and the extension is linear and of the same norm: $T \in B(S, W)$, $S \subset V$, $\exists! \tilde{T} \in B(\overline{S}, W)$: $\tilde T|_S = T$, $\|\tilde{T}\| = \|T\|$.

Hahn-Banach Theorem for normed spaces: Every bounded linear functional on a subspace of a normed space has a linear extension to the space with the same norm: $\omega \in S^\#$ then $\exists \tilde\omega \in V^\#$, $\|\tilde\omega\| = \|\omega\|$.

For every nonzero vector in a normed space there is a unit-norm linear functional that maps the vector to its norm: $\forall v \in V \setminus \{0\}$, $\exists \omega \in V^\#$, $\|\omega\| = 1$, $\omega v = \|v\|$. A vector in a normed space is zero if all bounded linear functionals on the space equals zero at the vector.

### Uniform Boundedness Theorem

Uniform Boundedness Theorem [@Banach and Steinhaus, 1927]: If a sequence of bounded linear operators between Banach spaces is point-wise bounded, then their operator norms are also bounded: $(T_n)_{n \in \mathbb{N}} \subset B(\widehat V, \widehat W)$, $\{(\|T_n v\|)_{n \in \mathbb{N}} : v \in \widehat V\} \subset l^\infty$, then $(\|T_n\|)_{n \in \mathbb{N}} \in l^\infty$.

The polynomial algebra over a field is not Banach when endowed with the norm that equals the maximum absolute coefficient. $(\mathbb{F} \left[x\right], \|\cdot\|)$, $\|\sum_{k=0}^n a_k x^k\| := \max_k |a_k|$.

Uniformly operator convergent... Strongly operator convergent... Weakly operator convergent... Strong convergence of a sequence of functionals... Weak* convergence of a sequence of functionals... Summability method... Summable sequence... Matrix method... Regular matrix method or permanent matrix method... Numerical integration process... Convergent numerical integration...

### Open Mapping Theorem and Closed Graph Theorem

Open Mapping Theorem or Bounded Inverse Theorem [@Banach1929]: Every surjective bounded linear operator between Banach spaces is an open mapping; hence every bijective bounded linear operator between Banach spaces has a continuous and thus bounded inverse: $T \in B(\widehat V, \widehat W)$ then $T^{-1} \in B(\widehat W, \widehat V)$.

Closed Graph Theorem [@Banach1929]: Every closed linear operator between Banach spaces is bounded: $T \in \mathcal{L}(\widehat V, \widehat W)$, $\Gamma(T) \in \mathcal{T}^∗_{V \times W}$, then $T \in B(\widehat V, \widehat W)$.

## Multilinear Operator

Multilinear operator $F: \prod_{i=1}^k V_i \mapsto W$ from a Cartesian product of vector spaces to a vector space, all over the same field, is a map that is a linear operator in each variable: $\forall j \in \{i\}_{i=1}^k$, $\forall v \in V$, $F|_{v-v_j+V_j} \in \mathcal{L}(V_j, W)$. For example, the determinant $\det$ is a multilinear functional of n vectors of an n-dimensional inner product space, which gives the oriented volume of the parallelepiped determined by these vectors. Multilinear operator space $\mathcal{L}(\prod_{i=1}^k V_i, W)$ is the vector space consisting of the set of all multilinear operators from $\prod_{i=1}^k V_i$ to $W$ and pointwise addition and scalar multiplication. Bilinear operator is a multilinear operator of two variables. For example, the inner product $(\cdot,\cdot)$ of a Euclidean space is a bilinear functional, which defines lengths of vectors and angles between them.

Operator norm of a multilinear operator from a Cartesian product of normed spaces to a normed space is the supremum of norms of the image of the product of unit balls: $\|F\| = \sup_{\|v_i\| \le 1} \|F(v_i)_{i=1}^k\|$; or equivalently, $\|F\| = \sup_{v_i \in V_i} \frac{\|F(v_i)_{i=1}^k\|}{\prod_{i=1}^k \|v_i\|}$. Space of bounded multilinear operators $B(\prod_{i=1}^k V_i,W)$ between normed spaces is the vector space of linear operators endowed with the operator norm: $B(\prod_{i=1}^k V_i, W) = (\mathcal{L}(\prod_{i=1}^k V_i, W), \|\cdot\|)$. A multilinear operator on normed spaces is continuous if and only if it is bounded. The norm of a symmetric multilinear operator on inner product spaces is determined by its action on tuples of the same vectors: if $\forall \pi \in S(k)$, $F \circ \pi = F$, then $\|F\| = \sup_{\|v\| \le 1} \|F(v)_{i=1}^k\|$.

k-times differentiable mapping between (open subsets of) Banach spaces is a (k-1)-times differentiable mapping whose (k-1)-th derivative is differentiable: $f \in C^{k-1}(U, W)$, $U \subset V$, $D^{k-1} f$ is differentiable. k-th derivative $D^k f$ of a k times differentiable mapping is a symmetric k-linear map at each point: $D^k f: U \mapsto \mathcal{L}(U^k, W)$, $\forall x \in A$, $\forall \pi \in S(k)$, $(D^k f)_x \circ \pi = (D^k f)_x$. If the domain is an open subset of the Euclidean n-space, partial derivatives of order k can be defined as ususal, and the k-th derivative can be written as: $[D^k f(x)]_I = \frac{\partial^k f(x)}{\partial x^I}$, where multi-index $I \in n^k$. Taylor's Formula: Every k-times continuously differentiable map between star-shaped subsets of Banach spaces can be written as the sum of multilinear maps and a k-th order term w.r.t. the center: $f \in C^k(U, W)$, $\forall [x, x + v] \subset U$, $f(x + v) = \sum_{i=0}^{k-1} \frac{(D^i f)_x}{i!}(v)_{j \in i} + g(v)_{j \in k}$, where $g = \int_0^1 \frac{(1 - s)^{k-1}}{(k-1)!} (D^k f)_{x + s v}~\text{d}s$; in particular, $\forall \varepsilon > 0$, $\exists \delta > 0$: $\forall \|v\| \le \delta$, $\left\|\left(g - \frac{(D^k f)_x}{k!}\right)(v)_{j \in k}\right\| \le \varepsilon \|v\|^k$.

## Approximation Theory

Approximation theory of functions is concerned with approximating functions of a certain kind by functions of another kind that is perhaps simpler. For example, approximating continuous real functions on an interval by polynomials, and approximating smooth real functions by its Taylor series. Such functions typically form vector spaces and we approximate from within a subspace, then we can talk about distance by endowing a norm to the space; the choice of the norm depends on the purpose.

Best approximation of a vector in a normed space on a subspace is a vector in the subspace that is closest to the point: $S \subset X$, $x \in X$, $x_0 \in S$: $d(x, x_0) = d(x, S)$. Every vector in a normed space has a best approximation in every finite-dimensional subspace: $\dim S < \infty$ then $\forall x \in X$, $d(x, S) = \min_{y \in S} d(x, y)$. Best approximations of a vector in a normed space on a subspace, if exist, form a convex subset: $\text{conv}(S_0) = S_0$ where $S_0 = \arg\min_{y \in S} d(x, y)$. Every vector in a strictly convex normed space has at most one best approximation on a subspace. Uniform approximation or Chebyshev approximation is best approximation in the uniform norm. Least squares approximation is best approximation in an $L^2$ norm. Other commonly used norms for best approximation include $L^1$ norm and (matrix) nuclear norm.

### Uniform Approximation

The uniform-normed space $(C([a, b]), \|\cdot\|_\infty)$ of continuous real functions on an interval is not strictly convex, so best approximations may not be unique on some subspaces. Haar condition on an n-dimensional subspace of a continuous real function space on an interval is the condition that every nonzero vector in the subspace has at most n-1 zeros: $\sup\{|f^{-1}(0)| : f \in S, f \ne 0\} \le \dim S - 1$; or equivalently, every basis of the subspace discretized at n distinct points is a basis for the Euclidean n-space: $\text{span}(f_i)_{i=1}^n = S$ then $\text{span}((f_i(x_j))_{j=1}^n)_{i=1}^n = \mathbb{R}^n$. Haar Uniqueness Theorem for best uniform approximation: Every vector in the uniform-normed space of continuous real functions on an interval has a unique best approximation in a finite-dimensional subspace if and only if the subspace satisfies the Haar condition. Subspaces $\text{Span}(x^k)_{k=0}^n$ of all polynomials of degree no greater than a given number satisfy the Haar condition, and thus continuous real functions on intervals have unique best uniform approximations on them. Chebyshev polynomial $T_n(x)$ of the first kind of order n is the polynomial defined by $T_n(x) = \cos(n \arccos x)$, $x \in [-1, 1]$. Here "T" is for "Tchebichef", another transliteration besides "Chebyshev". Chebyshev polynomial $U_n(x)$ of the second kind of order n is the polynomial defined by $U_n(x) = \sin(n \arccos x)$, $x \in [-1, 1]$.

Weierstrass Approximation Theorem for polynomials: The set of all real polynomials is dense in the uniform-normed space of continuous real functions on an interval: $\overline{\text{Span}(x^n)_{n \in \mathbb{N}}} = (C([a, b]), \|\cdot\|_\infty)$. Hence every continuous real function on an interval can be uniformly approximated by a sequence of polynomials: $\forall f \in C[a, b]$, $\exists (p_n) \subset P$: $\lim_{n \to \infty} \max |f - p_n| = 0$.

Müntz Approximation Theorem: A set of power functions is dense in the uniform-normed space of continuous real functions on an interval, where the exponents is an increasing sequence of real numbers from zero to infinity, if and only if the reciprocals of the positive terms sum to infinity: $a_0 = 0$, $a_n < a_{n+1}$, $\lim_{n \to \infty} a_n = \infty$, then $\overline{\text{Span}(x^{a_n})_{n \in \mathbb{N}}} = (C([a, b]), \|\cdot\|_\infty)$ iff $\sum_{n=1}^\infty 1 / a_n = \infty$.

Stone-Weierstrass Theorem: A subalgebra of continuous real-valued functions on a compact Hausdorff space is dense in the algebra endowed with the uniform norm, if it separates points in the space and contains constant functions: $(A, (+, \cdot_{\mathbb{R}}, ×)) \subset C(K)$, $1 \in A$, $\mathcal{T}\{f^{-1}(U) : f \in A, U \in \mathcal{T}_{\mathbb{R}}\} = \mathcal{T}_K$, then $\bar{A} = (C(K), \|\cdot\|_\infty)$. Stone-Weierstrass Theorem generalizes the Weierstrass Approximation Theorem for polynomials.

Weierstrass Approximation Theorem for trigonometric polynomials: Every continuous periodic function is the uniform limit of trigonometric polynomials.

## Spectral Theory

Spectral theory of operators is concerned with general properties of the inverse operators of certain operators, and their relations to the original operators. Spectral theory of operators on finite-dimensional vector spaces is essentially matrix eigenvalue theory.

Point spectrum... Eigenvalue of a linear transformation is an element of its point spectrum. Continuous spectrum... Residual spectrum...

Eigenvector or eigenfunction...

Spectral representation... Spectrum... Spectral resolution...

## Misc Theorems

Equicontinuous sequence of continuous real functions on an interval is a uniformly continuous sequence: $(f_i) \subset C[a, b]$, $\forall \varepsilon > 0$, $\exists \delta > 0$: $\forall i \in \mathbb{N}$, $\forall x \in [a, b]$, $f_i(B_\delta(x)) \subset B_\varepsilon(f_i(x))$. Arzela-Ascoli Theorem: Every bounded equicontinuous sequence in the normed space $(C[a, b], \|\cdot\|_\infty)$ of continuous real functions on an interval with the uniform norm has a convergent subsequence.

Bolzano-Weierstrass theorem: Every bounded sequence of real numbers has a convergent subsequence.

## References

• [@Kreyszig1978] Introductory Functional Analysis with Applications.