Normed Spaces and Banach Spaces

A normed space is a vector space $X$ over $\mathbb{R}$ or $\mathbb{C}$ together with a norm $\|\cdot\| : X \to [0, \infty)$ satisfying:

1. $\|x\| = 0 \iff x = 0$ (positive definiteness).
2. $\|\alpha x\| = |\alpha| \cdot \|x\|$ for all scalars $\alpha$ (homogeneity).
3. $\|x + y\| \leq \|x\| + \|y\|$ (triangle inequality).

A norm induces a metric $d(x, y) = \|x - y\|$, making $X$ a metric space.

Example 1. $(C[a, b], \|\cdot\|_\infty)$ with $\|f\|_\infty = \sup_{x \in [a,b]} |f(x)|$.

Example 2. $(C[a, b], \|\cdot\|_1)$ with $\|f\|_1 = \int_a^b |f(x)|\, dx$. This norm is weaker: convergence in $\|\cdot\|_1$ does not imply pointwise convergence.

Example 3. $\ell^p = \{(x_n) : \sum |x_n|^p < \infty\}$ with $\|x\|_p = (\sum |x_n|^p)^{1/p}$ for $1 \leq p < \infty$.

Example 4. $\ell^\infty = \{(x_n) : \sup_n |x_n| < \infty\}$ with $\|x\|_\infty = \sup_n |x_n|$.

A Banach space is a complete normed space (every Cauchy sequence converges).

Theorem 1.1. $\ell^p$ is a Banach space for $1 \leq p \leq \infty$.

Theorem 1.2. $L^p(\mu)$ is a Banach space for $1 \leq p \leq \infty$.

Theorem 1.3. $(C[a, b], \|\cdot\|_\infty)$ is a Banach space, but $(C[a, b], \|\cdot\|_1)$ is not (it is not complete: the limit of continuous functions in $L^1$-norm may be discontinuous).

Theorem 1.4. All norms on a finite-dimensional vector space are equivalent.

Corollary 1.5. Every finite-dimensional normed space is a Banach space.

Theorem 1.6 (Riesz’s Lemma). Let $X$ be a normed space and $Y$ a proper closed subspace. For every $0 < \theta < 1$, there exists $x \in X$ with $\|x\| = 1$ and $d(x, Y) \geq \theta$.

Corollary 1.7. The closed unit ball of a normed space is compact if and only if the space is finite-dimensional.

Let $X$ be a normed space and $Y \subseteq X$ a closed subspace. The quotient space $X / Y$ consists of equivalence classes $[x] = x + Y$ with the quotient norm:

$\|[x]\|_{X/Y} = \inf_{y \in Y} \|x - y\|$

Theorem 1.8. If $X$ is a Banach space and $Y$ is a closed subspace, then $X/Y$ is a Banach space.

Proposition 1.9. The quotient map $\pi : X \to X/Y$, $\pi(x) = [x]$, is a bounded linear operator with $\|\pi\| = 1$.

The dual space $X^*$ of a normed space $X$ is the space of all bounded linear functionals $f : X \to \mathbb{F}$, equipped with the operator norm:

$\|f\| = \sup_{\|x\| \leq 1} |f(x)|$

Theorem 1.10. The dual space $X^*$ is always a Banach space, regardless of whether $X$ is complete.

Examples of dual spaces:

• $(\ell^p)^* \cong \ell^q$ where $1/p + 1/q = 1$ for $1 \leq p < \infty$.
• $(c_0)^* \cong \ell^1$, where $c_0$ is the space of sequences converging to $0$.
• $(L^p(\mu))^* \cong L^q(\mu)$ for $1 \leq p < \infty$ and $1/p + 1/q = 1$.

Theorem 1.11. Every normed space $X$ has a completion: a Banach space $\tilde{X}$ and an isometric embedding $i : X \to \tilde{X}$ with dense image. The completion is unique up to isometric isomorphism.

Proof sketch. Take the set of Cauchy sequences in $X$, modulo the equivalence relation $(x_n) \sim (y_n)$ if $\|x_n - y_n\| \to 0$. Define $\tilde{X}$ as this set with the norm $\|[(x_n)]\| = \lim_{n\to\infty} \|x_n\|$. The map $i(x) = [(x, x, x, \ldots)]$ is an isometric embedding. $\blacksquare$

Example. The completion of $(C[a, b], \|\cdot\|_1)$ is $L^1[a, b]$.

Infinite-dimensional normed spaces have properties that contrast sharply with finite-dimensional ones:

• The closed unit ball is not compact (Riesz’s lemma).
• There exist discontinuous linear operators (requires the axiom of choice).
• Not every linear subspace is closed.
• The weak topology differs from the norm topology.

Theorem 1.12 (Hölder’s Inequality). For $1 \leq p, q \leq \infty$ with $1/p + 1/q = 1$:

$\sum_{n=1}^\infty |x_n y_n| \leq \|x\|_p \|y\|_q$

Theorem 1.13 (Minkowski’s Inequality). For $1 \leq p \leq \infty$:

$\|x + y\|_p \leq \|x\|_p + \|y\|_p$

These inequalities prove that $\ell^p$ and $L^p$ are normed spaces.

Problem 1. Show that $C[a, b]$ with $\|f\|_1 = \int_a^b |f(x)|\, dx$ is not complete.

Solution. Consider $f_n(x) = \begin{cases} 0 & a \leq x \leq (a+b)/2 - 1/n \\ \text{linear} & \text{in the transition} \\ 1 & (a+b)/2 + 1/n \leq x \leq b \end{cases}$. This is a Cauchy sequence in $\|\cdot\|_1$ but converges to the discontinuous step function. $\blacksquare$

Problem 2. Prove that $\ell^p \subset \ell^q$ for $1 \leq p < q \leq \infty$.

Problem 3. Show that $\|x\|_\infty = \lim_{p \to \infty} \|x\|_p$ for $x \in \ell^p \cap \ell^\infty$.

Problem 4. Prove that the dual of $c_0$ is $\ell^1$.

A normed space $X$ carries the weak topology $\sigma(X, X^*)$, the coarsest topology making all $f \in X^*$ continuous. A sequence converges weakly ($x_n \rightharpoonup x$) if $f(x_n) \to f(x)$ for every $f \in X^*$.

The dual space $X^*$ carries the weak- topology* $\sigma(X^*, X)$, the coarsest topology making all evaluation maps $x \mapsto f(x)$ continuous.

Theorem 1.14 (Banach-Alaoglu). The closed unit ball of $X^*$ is compact in the weak-* topology.

A normed space is separable if it contains a countable dense subset.

Examples: $\ell^p$ is separable for $1 \leq p < \infty$. $\ell^\infty$ is not separable. $C[a, b]$ is separable (polynomials with rational coefficients are dense).

Theorem 1.15. If $X^*$ is separable, then $X$ is separable. The converse does not hold: $\ell^1$ is separable but $(\ell^1)^* \cong \ell^\infty$ is not.

A Banach space $X$ is reflexive if the natural embedding $J : X \to X^{**}$ defined by $J(x)(f) = f(x)$ is surjective.

Examples: $\ell^p$ is reflexive for $1 < p < \infty$. $\ell^1$ and $\ell^\infty$ are not reflexive. Every finite-dimensional space is reflexive.

Theorem 1.16. A Banach space is reflexive if and only if its closed unit ball is weakly compact.

Problem 5. Show that $c_0$ (sequences converging to 0 with sup norm) is not reflexive.

Problem 6. Prove that $\ell^p$ for $1 < p < \infty$ is reflexive using the fact that $(\ell^p)^{**} \cong \ell^p$ via the natural embedding.

• $L^p$ Spaces: Provides concrete examples of Banach spaces that are fundamental throughout analysis, with the completeness result proved here.

• Inner Product Spaces and Hilbert Spaces: Specialises the normed space theory to spaces where the norm comes from an inner product, enabling orthogonal projections.

• Bounded Linear Operators: Develops the theory of continuous linear maps between normed spaces, building on the dual space concept.

• Quantum Mechanics

• Graph Theory

• Classical Mechanics

• Electromagnetism

flowchart TD
    A[1_Normed Spaces And Banach Spaces] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

Functional analysis extends linear algebra to infinite dimensions. A normed space is a vector space where you can measure the “size” of vectors, and a Banach space is one where Cauchy sequences converge — the space has no “holes.” Think of it as doing linear algebra with functions instead of finite arrays. The dual space contains all continuous linear functionals: machines that take in a vector and return a number. Boundedness and continuity coincide for linear maps, a fact that fails in nonlinear settings. The key challenge of infinite dimensions is that bounded sequences need not have convergent subsequences — compactness becomes a rare and precious property, requiring special conditions like reflexivity.

Mistake 1: Confusing a norm with a metric A norm $\|\cdot\|$ on a vector space induces a metric $d(x, y) = \|x - y\|$, but not every metric comes from a norm. The discrete metric $d(x, y) = 1$ for $x \neq y$ is not induced by any norm because norms are homogeneous ($\|\alpha x\| = |\alpha|\|x\|$), which the discrete metric violates. Always verify that a metric is translation-invariant and homogeneous before assuming it comes from a norm.

Mistake 2: Assuming bounded linear operators are automatically continuous In the context of linear operators, boundedness and continuity are equivalent for linear maps between normed spaces. However, students sometimes confuse bounded linear functionals with bounded sets. A bounded linear functional $f \in X^*$ satisfies $\|f(x)\| \leq C\|x\|$ for some constant $C$, not that $f(X)$ is a bounded set.

Mistake 3: Forgetting that $(C[a,b], \|\cdot\|_1)$ is not complete The space of continuous functions with the $L^1$-norm is not a Banach space because Cauchy sequences can converge to discontinuous functions. Only $(C[a,b], \|\cdot\|_\infty)$ is complete. When completeness is required, always verify the norm, not just the underlying set.