Measure Integral and Real Analysis (3)

Banach Spaces

Metric Spaces

Definition 6.1: Metric Space

A metric oon a nonempty set \(V\) is a function \(d: V \times V \rightarrow [0, \infty)\) s.t:

• \(d(f, f) = 0, \; \forall f \in V\).
• If \(f, g \in V\) and \(d(f, g) = 0\), then \(f = g\).
• \(d(f, g) = d(g, f), \; \forall f, g \in V\).
• \(d(f, h) \leq d(f, g) + f(g, h), \; \forall f, g, h \in V\). (Triangle inequality)

A metric space is a pair \((V, d)\), where \(V\) is a nonempty set adn \(d\) is a metric on \(V\).

Definition 6.3: Open Ball; \(B(f, r)\); Closed Ball \(\bar{B}(f, r)\)

Suppose \((V, d)\) is a metric space, \(f \in V\) and \(r > 0\):

• The open ball centered at \(f\) with radius \(r\) is denoted \(B(f, r)\) and is defined by: \[B(f, r) = \{g \in V: d(f, g) < r\}\]
• The closed ball centered at \(f\) with radius \(r\) is denoted \(\bar{B}(f, r)\) and is defined by: \[\bar{B}(f, r) = \{g \in V: d(f, g) \leq r\}\]

Definition 6.4: Open

A subset \(G\) of a metric space \(V\) is called open if for every \(f \in G\), there exists \(r > 0\) s.t \(B(f, r) \subseteq G\).

Theorem 6.5: Open balls are Open

Suppose \(V\) is a metric space, \(f \in V\), and \(r > 0\). Then \(B(f, r)\) is an open subset of \(V\).

Proof of Theorem 6.5:

Suppose \(g \in B(f, r)\). We need to show there exists an open ball centered at \(g\) with radius \(k\) s.t \(B(g, k) \subseteq B(f, r)\). Let \(k = r - d(f, g)\), and \(h \in B(g, r - d(f, g))\), then:

\[d(f, h) \leq d(f, g) + d(g, h) < d(f, g) + r - d(f, g) = r\]

Thus, \(h \in B(f, r), \; \forall h \in B(g, k)\), so \(B(f, r)\) is open.

Definition 6.6: Closed

A subset of a metric space \(V\) is called closed if its complement in \(V\) is open.

Definition 6.7: Closure; \(\bar{E}\)

Suppose \(V\) is a metric space and \(E \subseteq V\). The closure of \(E\), denoted \(\bar{E}\), is defined by:

\[\bar{E} = \{g \in V: B(g, \epsilon) \cap E \neq \emptyset \; \forall \epsilon > 0\}\]

Definition 6.8: Limit in Metric Space; \(\lim_{k \rightarrow \infty} f_k\)

Suppose \((V, d)\) is a metric space, \(f_1, f_2, ...\) is a sequence in \(V\), and \(f \in V\). Then:

\[\lim_{k\rightarrow \infty} f_k = f\]

means

\[\lim_{k \rightarrow \infty} d(f, f_k) = 0\]

Theorem 6.9: Closure

Suppose \(V\) is a metric space and \(E \subseteq V\). Then:

1. \(\bar{E} = \{g \in V: \text{there exist $f_1, f_2, ...$ in $E$ s.t $\lim_{k \rightarrow \infty} f_k = g$}\}\)
2. \(\bar{E}\) is the intersection of all closed subsets of \(V\) that contains \(E\).
3. \(\bar{E}\) is a closed subset of \(V\).
4. \(E\) is closed if and ony if \(\bar{E} = E\).
5. \(E\) is closed if and only if \(E\) contains the limit of every convergent sequence of elements of \(E\).

Definition 6.10: Continuous

Suppose \((V, d_v)\) and \((W, d_w)\) are metric spaces and \(T: V \rightarrow W\) is a function.

• For \(f \in V\), the function \(T\) is called continuous at \(f\) if for every \(\epsilon > 0\), there exists \(\delta > 0\) s.t: \[d_w (T(f), T(g)) < \epsilon\]

  for all \(g \in V\) amd \(d_v(f, g) < \epsilon\).

• The function \(T\) is called continuous if \(T\) is continuous at \(f\) for every \(f \in V\).

Theorem 6.11: Equivalent Conditions for Continuity

Suppose \(V, W\) are metric spaces and \(T: V \rightarrow W\) is a function. Then the following are equivalent:

1. \(T\) is continuous.
2. \(\lim_{k \rightarrow \infty} f_k = f \in V \implies \lim_{k \rightarrow \infty} T(f_k) = T(f) \in W\).
3. \(T^{-1} (G)\) is an open subset of \(V\) for every open set \(G \subseteq W\).
4. \(T^{-1} (F)\) is a closed subset of \(V\) for every closed set \(F \subseteq W\).

Definition 6.12: Cauchy Sequence

A sequence \(f_1, f_2, ...\) in a metric space \((V, d)\) is called a Cauchy sequence if for every \(\epsilon > 0\), there exists \(n \in \mathbb{Z}^+\) s.t \(d(f_i, f_k) < \epsilon\) for all integers \(j \geq n\) and \(k \geq n\).

Theorem 6.13: Every Convergent Sequence is a Cauchy Sequence

Every convergent sequence in a metric space is a Cauchy seqeunce.

Definition 6.14: Complete Metric Space

A metric space \(V\) is called complete if every Cauchy sequence in \(V\) converges to some element of \(V\).

Theorem 6.16: Connection Between Complete and Closed

1. A complete subset of a metric space is closed.
2. A closed subset of a complete metric space is complete.

Proof of Theorem 6.16:

To prove a, assume \(U \subseteq V\) is a complete subset of a metric space \((V, d)\), then for convergent sequence \(f_1, f_2, ... \in U\), we have the limit \(f \in V\), which implies the set is closed.

To prove b, assume \(U \subseteq V\) is a closed subset of a complete metric space \(V\), then for cauchy sequence \(f_1, f_2, ... \in U\), we also have \(f_1, f_2, ... \in V\). Since \(V\) is complete, \(f_1, f_2, ..\) has a limit \(f \in V\), since \(U\) is closed, \(f \in U\).

Vector Spaces

If \(z_1, z_2, ...\) is a sequence of complex numbers and \(L \in \mathbb{C}\), then:

\[\lim_{k \rightarrow} z_k = L \implies \lim_{k \rightarrow \infty} |z_k - L| = 0\]

IFF

\[\lim_{k\rightarrow \infty} Re(z_k) = Re(L), \quad \quad \lim_{k\rightarrow \infty} Im(z_k) = Im(L)\]

Definition 6.19: Measurable Complex-Valued Function

Suppose \((X, S)\) is a measurable space. A function \(f: X \rightarrow \mathbb{C}\) is called \(S\)-measurable if \(Re(f)\) and \(Im(f)\) are both \(S\)-measurable functions.

The following are equivalent:

1. \(f\) is \(S\)-measurable.
2. \(f^{-1}(G) \in S\) for every open set \(G \in \mathbb{R}^2\).
3. \(f^{-1}(B) \in S\) for every Borel set \(B \in \mathbb{B}_2\).

Theorem 6.20: \(|f|^p\) is Measurable If \(f\) is Measurable

Suppose \((X, S)\) is a measurable space, \(f: X \rightarrow \mathbb{C}\) is an \(S\)-measurable function, and \(0 < p < \infty\). Then \(|f|^p\) is an \(S\)-measurable function.

Definition 6.21: Integral of a Complex-valued Function; \(\int f d\mu\)

Suppose \((X, S, \mu)\) is a measure space and \(f: X \rightarrow \mathbb{C}\) is an \(S\)-mearuable function with \(\int |f| d\mu < \infty\) (\(f \in L^1(\mu)\)) Then \(\int f d\mu\) is defined by:

\[\int f d\mu = \int (Re(f)) d\mu + i\int (Im (f)) d\mu\]

And

\[\int (f + g) d\mu = \int f d\mu + \int g d\mu\]

\[\int \alpha f d\mu = \alpha \int f d\mu\]

where \(f, g\) are S-measurable complex valued functions s.t the integrals are defined, and \(\alpha \in \mathbb{C}\).

Theorem 6.22: Bound on the Absolute Value of an Integral

Suppose \((X, S, \mu)\) is a measure space and \(f: X \rightarrow \mathbb{C}\) is an \(S\)-measurable function s.t \(\int |f| d\mu < \infty\). Then:

\[|\int f d\mu| \leq \int |f| d\mu\]

Theorem 6.24: Integral of Complex Conjugate of a Function

\[\int \bar{f} d\mu = \overline{\int f d\mu}\]

for every measure \(\mu\) and every \(f \in L^1 (\mu)\)

Normed Vector Space

Definition 6.32: \(\mathbb{F}^X\)

\(\mathbb{F}^X\) denotes the space of all functions that maps from \(X \rightarrow \mathbb{F}\), this is a vector space.

Definition 6.33: Norm; Normed Vector Space

A norm on a vector space \(V\) over $ $ is a function \(\|\cdot\|: V \rightarrow [0, \infty)\) s.t:

• \(\|f\| = 0\) IFF \(f = 0\) (positive definite)
• \(\|\alpha f\| = |\alpha| \|f\|\) (Homogeneity)
• \(\|f + g\| \leq \|f\| + \|g\|, \; \forall f,g \in V\) (Triangle Inequality)

A normed vector space is a pair \((V, \|\cdot\|)\), where \(V\) is a vector space and \(\|\cdot\|\) is a norm on \(V\).

Definition 6.34: Norm Equivalence

We say two norms \(F, G\) on \(V\) are equivalent if there is a \(\lambda < \infty\) s.t:

\[\lambda^{-1}G(x) \leq F(x) \leq \lambda G(x)\]

for all \(x \in V\)

Theorem 6.35: Any Two Norms on a Finite-Dimensional Space are Equivalent

Any two norms on a finite-dimensional vector space are equivalent.

Theorem 6.36: Normed Vector Spaces are Metric Spaces

Suppose \((V, \|\cdot\|)\) is a normed vector space. Define \(d: V \times V \rightarrow [0, \infty)\) by:

\[d(f, g) = \|f - g\|\]

Then \(d\) is a metric on \(V\).

So in the context of normed vector space:

\[\lim_{k\rightarrow \infty} f_k = f \Longleftrightarrow \lim_{k \rightarrow \infty} \|f_k - f\| = 0\]

A sequence \(f_1, f_2, ...\) in a normed vector space \((V, \|\cdot\|)\) is a Cauchy sequence if for every \(\epsilon > 0\), there exits \(n \in \mathbb{Z}^+\) s.t \(\forall k, j \in \mathbb{Z}^+ \geq n\), \(\|f_k - f_j\| < \epsilon\).

Definition 6.37: Banach Space

A complete normed vector space is called a Banach space.

In other words, a normed vector space is a Banach space, if every cauchy sequence in \(V\) converges to some element of \(V\).

Definition 6.40: Infinite Sum in a Normed Vector Space

Suppose \(g_1, g_2, ...\) is a sequence in a normed vector space \(V\). Then \(\sum^{\infty}_{k=1} g_k\) is defined by:

\[\sum^{\infty}_{k=1} g_k = \lim_{n \rightarrow \infty} \sum^n_{k=1} g_k\]

If this limit exists, in which case the infinite series is said to converge.

Theorem 6.41:

Suppose \(V\) is a normed vector space. Then \(V\) is a banach space IFF \(\sum^{\infty}_{k=1} g_k\) converges for every sequence \(g_1, g_2, ...\) in \(V\) s.t \(\sum^{\infty}_{k=1} \|g_k\| < \epsilon\)

Definition 6.43: Bounded Linear Map; \(\|T\|\); \(B(V, W)\)

Suppose \(V, W\) are normed vector spaces over the same field \(\mathbb{F}\) and \(T: V \rightarrow W\) is a linear map.

• The norm of \(T\), denoted \(\|T\|\), is defined by: \[\|T\| = \sup\{\|T(f)\|: f\in V, \; \|f\| \leq 1\}\]
• \(T\) is called bounded if \(\|T\| < \infty\).
• The set of bounded linear maps from \(V\) to \(W\) is denoted \(B(V, W)\).

Theorem 6.46: \(\|\cdot\|\) is a Norm on \(B(V, W)\)

Suppose \(V, W\) are normed vector spaces, \(\|\cdot\|\) is defined above. Then \(\forall S, T \in B(V, W), \alpha \in \mathbb{F}\):

• \(\|S + T\| \leq \|S\| + \|T\|\)
• \(\|\alpha T\| = |\alpha|\|T\|\) Furthermore, the function \(\|\cdot\|\) is a norm on \(B(V, W)\)

Theorem 6.47(1): Equivalence Conditions For Norm of Linear Maps

All Definitions below are equivalent for normed vector spaces \(V, W\), \(V \neq \{0\}\), \(T: V \rightarrow W\):

1. \(\|T\| = \sup\{\|T(f)\|: f\in V, \; \|f\| \leq 1\}\)
2. \(\|T\| = \sup\{\|T(f)\|: f\in V, \; \|f\| = 1\}\)
3. \(\|T\| = \inf \{c \in [0, \infty): \|T(f)\| \leq c\|f\|, \forall f \in V\}\)
4. \(\|T\| = \sup\{\frac{\|T(f)\|}{\|f\|}: f \in V, \; f \neq 0\}\)

This conditions implies that:

\[\|T(f)\| \leq \|T\|\|f\|, \; \forall f\in V\]

Theorem 6.47: \(B(V, W)\) is a Banach Space IF \(W\) is a Banach Space

Suppose \(V\) is a normed vector space and \(W\) is a Banach Space. Then \(B(V, W)\) is a Banach space.

Theorem 6.48: Continuity is Equivalent to Boundedness for Linear Maps

A linear map from one normed vector space to another normed vector space is continuous IFF its bounded.

Theorem 6.49: A Linear Map from a Finite-Dimensional Space is Always Continuous

A linear map from a finite-dimensional space is always continuous.

Linear Functions

Theorem 6.52: Bounded Linear Functionals

Suppose \(V\) is a normed vector space and \(\varphi: V \rightarrow \mathbb{F}\) is a linear functional that is not identically 0. Then the following are equivalent:

1. \(\varphi\) is a bounded linear functional.
2. \(\varphi\) is a continuous linear functional.
3. \(\text{null } \varphi\) is a closed subspace of \(V\).
4. \(\overline{\text{null $\varphi$}} \neq V\).

Definition 6.53: Family

A family \(\{e_k\}_{k \in \Gamma}\) in a set \(V\) is a function \(e\) from a set \(\Gamma\) to \(V\) with the value of the function \(e\) at \(k \in \Gamma\) denoted by \(e_k\).

In other words, \(e: \Gamma \rightarrow V\) is defined as:

\[e(k) = e_k, \forall k \in \Gamma\]

The range of the function \(e\) is a subset of \(V\):

\[range(e) = \{e_k\}_{k \in \Gamma}\]

A subset \(\Gamma\) of a vector space \(V\), can be thought as a family in \(V\) by considering \(\{e_f\}_{f\in\Gamma}, e_f = f\).

We can think of \(\{e_k\}\) as a list.

Definition 6.54: Linearly independent; Span; Finite-dimensional; Basis

Suppose \(\{e_k\}_{k\in\Gamma}\) is a family in a vector space \(V\).

• \(\{e_k\}_{k\in\Gamma}\) is called linearly independent if there does not exist a finite nonempty subset \(\Omega\) of \(\Gamma\) and a family \(\{\alpha_j\}_{j \in \Omega}\) in \(F / \{0\}\) s.t: \[\sum_{j \in \Omega} \alpha_j e_j = 0\]
• The span of \(\{e_k\}_{k \in \Gamma}\) is denoted by \(span(\{e_k\}_{k \in \Gamma})\) and is defined to be the set of all sums of the form: \[\sum_{j \in \Omega} \alpha_j e_j\]
• A vector space \(V\) is called finite-dimensional if there exists a finite set \(\Gamma\) and a family \(\{e_k\}_{k \in \Gamma}\) in \(V\) s.t \(span(\{e_K\}_{k \in \Gamma}) = V\).
• A vector space is called infinite-dimensional if it is not finite-dimensional.
• A family in \(V\) is called basis (Hamel basis, which only finite sums are under consideration) of \(V\) if it is linearly independent and its span equals \(V\).

Definition 6.55: Maximal Element

Suppose \(A\) is a collection of subsets of a set \(V\). A set \(\Gamma \in A\) is called a maximal element of \(A\) if there does not exist \(\Gamma^\prime \in A\) s.t \(\Gamma \subset \Gamma^\prime\).

Theorem 6.57: Bases as Maximal Elements

Suppose \(V\) is a vector space. Then a subset of \(V\) is a basis of \(V\) IFF, it is a maximal element of the collection of linearly independent subsets of \(V\).

Definition 6.58: Chain

A collection \(C\) of subsets of a set \(V\) is called a chain if \(\Omega, \Gamma \in C \implies \Omega \subseteq \Gamma\) or \(\Gamma \subseteq \Omega\).

Theorem 6.60: Zorn's Lemma

Suppose \(V\) is a set and \(A\) is a collection of subsets of \(V\) with the property that the union of all the sets in \(C\) is in \(A\) for every chain \(C \subseteq A\). Then \(A\) contains a maximal element.

Theorem 6.61: Bases Exist

Every vector space has basis.

Theorem 6.62: Discontinuous Linear Functionals

Every infinite-dimensional normed vector space has a discontinuous linear functional.

Theorem 6.63: Extension Lemma

Suppose \(V\) is a real normed vector space, \(U\) is a subspace of \(V\), and \(\psi: U \rightarrow \mathbb{R}\) is a bounded linear functional. Suppose \(h \in V / U\). Then \(\psi\) can be extended to a bounded linear functional \(\varphi: U + \mathbb{R}h \rightarrow \mathbb{R}\) s.t \(\|\varphi\| = \|\psi\|\).

Where \(V / U = \{v + u: \forall v \in V, u \in U\}\) is a quotient space, and \(U + \mathbb{R}h = \{f + \alpha h: f \in U, \alpha \in \mathbb{R}\}\).

Definition 6.67: Graph

Suppose \(T: V \rightarrow W\) is a function from a set \(V\) to a set \(W\). Then the graph of \(T\) is denoted \(graph(T)\) and is the subset of \(V \times W\) defined by:

\[graph(T) = \{(f, T(f)) \in V \times W: f \in V\}\]

Formally, a function from a set \(V\) to a set \(W\) equals its graph.

Theorem 6.68: Function Properties in Terms of Graphs

Suppose \(V, W\) are normed vector spaces and \(T: V \rightarrow W\) is a function:

1. \(T\) is a linear map IFF \(graph(T)\) is a subspace of \(V \times W\).
2. Suppose \(U \subseteq V\) and \(S: U \rightarrow W\) is a function. Then \(T\) is an extension of \(S\) IFF \(graph(S) \subseteq graph(T)\).
3. If \(T: V \rightarrow W\) is a linear map and \(c \in [0, \infty)\), then \(\|T\| \leq c\) IFF \(\|g\| \leq c \|f\|\) for all \((f, g) \in graph(T)\).

Theorem 6.69: Hahn-Banach Theorem

Suppose \(V\) is a normed vector space, \(U\) is a subspace of \(V\), and \(\psi: U \rightarrow \mathbb{F}\) is a bounded linear functional. Then \(\psi\) can be extended to a bounded linear functional on \(V\) whose norm equals \(\|\psi\|\).

Definition 6.71: Dual Space; \(V^{\prime}\)

Suppose \(V\) is a normed vector space. THen the dual space of \(V\), denoted \(V^{\prime}\) is the normed vector space consisting of the bounded linear functional on \(V\). In other words, \(V^{\prime} = B(V, \mathbb{F})\)

Theorem 6.72: \(\|f\| = \max_{\|\varphi\| = 1}\{|\varphi(f)|: \varphi \in V^{\prime}\}\)

Suppose \(V\) is a normed vector space and \(f \in V / \{0\}\). Then there exists \(\varphi \in V^{\prime}\) s.t \(\|\varphi\| = 1\) and \(\|f\| = \varphi (f)\).

\(L^p\) Spaces

\(L^p(E)\) where \(E\) is a Borel or Lebesgue subset of \(\mathbb{R}\) and \(0 < p \leq \infty\), then \(L^p(E)\) means \(L^p(\lambda_E)\), where \(\lambda_E\) is the lebesgue measure restricted to Borel or Lebesgue subsets contained in \(E\).

\(L^p (\mu)\)

Definition 7.1: \(\|f\|_p\); Essential Supremum

Suppose that \((X, S, \mu)\) is a measure space, \(0 < p < \infty\), and \(f: X \rightarrow \mathbb{F}\) is \(S\)-measurable. Then the \(p\)-norm of \(f\) is denoted by \(\|f\|_p\) and is defined by:

\[\|f\|_p = (\int |f|^p d\mu)^{\frac{1}{p}}\]

Also, \(\|f\|_{\infty}\) which is called the essential supremum of \(f\), is defined by:

\[\|f\|_{\infty} = \inf\{t > 0: \mu(\{x \in X: |f(x)| > t\}) = 0\}\]

The terminology \(p\)-norm is convenient, even though it is not necessarily a norm, for example If there exists a nonempty set \(E \in S\) s.t \(\mu(E) = 0\), then \(\|\chi_E\|_p = 0\), even if \(\chi_E \neq 0\).

We can think of \(\|f\|_{\infty}\) as the smallest that you can make the supremum of \(|f|\) after modifications on sets of measure \(0\).

Definition 7.3: Lebesgue Space; \(L^p (\mu)\)

Suppose \((X, S, \mu)\) is a measure space and \(0 < p \leq \infty\). The Lebesgue space \(L^p(\mu)\), sometimes denoted \(L^p(X, S, \mu)\) is defined to be the set of \(S\)-measurable functions \(f: X \rightarrow \mathbb{F}\) s.t \(\|f\|_p < \infty\).

Example \(l^p\)

When \(\mu\) is counting measure on \(\mathbb{Z}^+\), the set \(L^p(\mu)\) is often denoted by \(l^p\). Thus if \(0 < p < \infty\), then: \[l^p = \{(a_1, a_2, ...)\}: \text{ each } a_k \in \mathbb{F}, \; \sum^{\infty}_{k=1} |a_k|^p < \infty\}\] \[l^\infty = \{(a_1, a_2, ...): \text{ each } a_k \in \mathbb{F}, \; \sup_{k \in \mathbb{Z}^+} |a_k| < \infty\}\]

Theorem 7.5: \(L^p(\mu)\) is a Vector Space

Suppose \((X, S, \mu)\) is a measure space and \(0 < p < \infty\). Then:

1. \[\|f + g\|^p_p \leq 2^p (\|f\|^p_p + \|g\|^p_p)\]

2. \[\|\alpha f\|_p = |\alpha| \|f\|_p\]

for all \(f, g \in L^p(\mu)\) and all \(\alpha \in \mathbb{F}\). Furthermore, with the usual operations of addition and scalar multiplication of functions, \(L^p(\mu)\) is a vector space.

Proof of Theorem 7.5:

Suppose \(f, g \in L^p (\mu)\). If \(x \in X\), then:

\[|f(x) - g(x)|^p \leq (|f(x)| + |g(x)|)^p \leq (2 \max{|f(x)|, |g(x)|}^p) \leq 2^p (|f(x)|^p, |g(x)|^p)\]

Take the integral over both sides we have:

\[\int |f(x) - g(x)|^p d\mu \leq \int 2^p (|f(x)|^p, |g(x)|^p) d\mu \implies \|f + g\|^p_p \leq 2^p (\|f\|^p_p + \|g\|^p_p)\]

From this we can see that \(f, g \in L^p(\mu) \implies \|f + g\|_p < \infty \in L^p(\mu)\)

The scalar multiplication is straight forward, since \(0 \in L^p(\mu)\) and \(L^p(\mu)\) is closed under scalar multiplication and addition, it is a vector space and subspace of \(\mathbb{F}^X\).

Definition 7.6: Dual Exponent; \(p^{\prime}\)

For \(1 \leq p \leq \infty\), the dual exponent of \(p\) is denoted by \(p^\prime\) and is the element of \([1, \infty]\) s.t

\[\frac{1}{p} + \frac{1}{p^{\prime}} = 1\]

Example 7.7

\[1^{\prime} = \infty, \quad \infty^{\prime} = 1, \quad 2^{\prime} = 2, \quad 4^{\prime} = \frac{4}{3}\]

Theorem 7.8: Young's Inequality

Suppose \(1 < p < \infty\). Then:

\[ab \leq \frac{a^p}{p} + \frac{b^{p^{\prime}}}{p^{\prime}}\]

Theorem 7.9: Holder's Inequality

Suppose \((X, S, \mu)\) is a measure space, \(1 \leq p \leq \infty\), and \(f, h: X \rightarrow \mathbb{F}\) are \(S\)-measurable. Then:

\[\|fh\|_1 \leq \|f\|_p\|h\|_{p^{\prime}}\]

Theorem 7.10: \(L^q(\mu) \subseteq L^p(\mu)\) IF \(\mu(X) < \infty\)

Suppose \((X, S, \mu)\) is a finite measure space and \(0 < p < q < \infty\). Then:

\[\|f\|_p \leq \mu(X)^{\frac{q - p}{pq}}\|f\|_q\]

for all \(f \in L^q(\mu)\). Furthermore, \(L^q(\mu) \subseteq L^p(\mu)\).

Theorem 7.12: Formula for \(\|f\|_p\)

Suppose \((X, S, \mu)\) is a measure space, \(1 \leq p < \infty\), and \(f \in L^p(\mu)\). Then:

\[\|f\|_p = \sup\{|\int fhd\mu|: h \in L^{p^{\prime}}(\mu), \|h\|_{p^{\prime}} \leq 1\}\]

This result hold for \(p = \infty\), if \(\mu\) is \(\sigma\)-finite measure.

Theorem 7.14: Minkowski's Inequality

Suppose \((X, S, \mu)\) is a measure space, \(1 \leq p \leq \infty\), and \(f, g \in L^p(\mu)\). Then:

\[\|f + g\|_p \leq \|f\|_p + \|g\|_p\]

\(\mathbf{L}^p(\mu)\)

Mathematicians often pretend that elements of \(L^p(\mu)\) are functions, where two functions are considered to be equal if they differ only on a set of \(\mu\)-measure \(0\).

Definition 7.15: \(Z(\mu)\); \(\tilde{f}\)

Suppose \((X, S, \mu)\) is a measure space and \(0 < p \leq \infty\).

• \(Z(\mu)\) denotes the set of \(S\)-measurable functions from \(X \rightarrow \mathbb{F}\) that equal \(0\) almost everywhere (zero everywhere except for the set with zero measure). So \(Z(\mu)\) is a subspace of \(L^p(\mu)\).
• For \(f \in L^p(\mu)\), let \(\tilde{f}\) be a subset of \(L^p(\mu)\) defined by: \[\tilde{f} = \{f + z: z \in Z(\mu)\}\]
• If \(f, F \in L^p(\mu)\), then \(\tilde{f} = \tilde{F}\) IFF \(f(x) = F(x)\) for almost every \(x \in X\).

Definition 7.16: \(\mathbf{L}^p (\mu)\)

Suppose \(\mu\) is a measure and \(0 < p \leq \infty\).

• Let \(\mathbf{L}^p(\mu)\) denote the collection of subsets of \(L^p(\mu)\) defined by: \[\mathbf{L}^p(\mu) = \{\tilde{f}: f \in L^p(\mu)\}\]
• For \(\tilde{f}, \tilde{g} \in \mathbf{L}^p(\mu)\) and \(\alpha \in \mathbb{F}\), define \(\tilde{f} + \tilde{g}\) and \(\alpha \tilde{f}\) by: \[\tilde{f} + \tilde{g} = \widetilde{(f + g)} \quad \quad \alpha \tilde{f} = \widetilde{(\alpha f)}\]
• \(\mathbf{L}^p(\mu) = L^p(\mu) / Z(\mu)\)
• We can think of elements of \(L^p (\mu)\) as equivalence classes of function in \(L^p(\mu)\), where two functions are equivalent if they agree almost everywhere.

Definition 7.17: \(\|\cdot\|_p\) on \(\mathbf{L}^p (\mu)\)

Suppose \(\mu\) is a measure and \(0 < p \leq \infty\). Define \(\|\cdot\|_p\) on \(\mathbf{L}^p (\mu)\) by:

\[\|\tilde{f}\|_p = \|f\|_p\]

for \(f \in L^p(\mu)\)

Definition 7.18: \(\mathbf{L}^p (\mu)\) is a Normed Vector Space

Suppose \((X, S, \mu)\) is a measure space and \(1 \leq p \leq \infty\). Then \(\mathbf{L}^p (\mu)\) is a vector space and \(\|\cdot\|_p\) is a norm on \(\mathbf{L}^p (\mu)\).

Theorem 7.18.1: \(l^p = L^p(\mu) = \mathbf{L}^p(\mu)\)

If \(\mu\) is counting measure on \(\mathbb{Z}^+\), then: \[l^p = L^p(\mu) = \mathbf{L}^p(\mu)\]

Because the counting measure has no sets of measure \(0\) other thant the empty set.

Definition 7.19: \(\mathbf{L}^p(E)\) for \(E \subseteq \mathbb{R}\)

If \(E\) is a Borel or Lebesgue measurable subset of \(\mathbb{R}\) and \(0 < p \leq \infty\), then \(\mathbf{L}^p(E)\) means \(\mathbf{L}^p(\lambda_E)\), where \(\lambda_E\) denotes Lebesgue measure \(\lambda\) restricted to the Borel or Lebesgue measurable subsets of \(\mathbb{R}\) that are contained in \(E\).

Theorem 7.20: Cauchy Sequences in \(L^p(\mu)\) Converges

Suppose \((X, S, \mu)\) is a measure space and \(1 \leq p \leq \infty\). Suppose \(f_1, f_2, ...\) is a sequence of functions in \(L^p(\mu)\) s.t for every \(\epsilon > 0\), there exists \(n \in \mathbb{Z}^+\) s.t: \[\|f_j - f_k\|_p < \epsilon\]

for all \(j \geq n\) and \(k \geq n\). Then there exists \(f \in L^p(\mu)\) s.t: \[\lim_{k \rightarrow \infty} \|f_k - f\|_p = 0\]

Theorem 7.23: Convergent Sequences in \(L^p\) have Pointwise Convergent Subsequences

Suppose \((X, S, \mu)\) is a measure space and \(1 \leq p \leq \infty\). Suppose \(f \in L^p(\mu)\) and \(f_1, f_2, ...\) is a sequence of functions in \(L^p(\mu)\) s.t \(\lim_{k\rightarrow \infty} \|f_k - f\|_{p} = 0\), then there exists a subsequence \(f_{k_1}, f_{k_2}, ....\) s.t: \[\lim_{m\rightarrow \infty} f_{k_m} (x) = f(x)\]

for almost every \(x \in X\).

Theorem 7.24: \(\mathbf{L}^p(\mu)\) is a Banach Space

Suppose \((X, S, \mu)\) is a measure space, and \(1 \leq p \leq \infty\). Then, \(\mathbf{L}^p(\mu)\) is a Banach Space.