The Central Idea

Metric spaces abstract the idea of distance. Normed spaces arise when the points being compared are vectors and their distance comes from a notion of vector magnitude.

If \(V\) is a vector space and \(\|\cdot\|\) is a norm on \(V\), then

\[ d(x,y)=\|x-y\| \]

defines a metric.

Thus every normed space is automatically a metric space, but not every metric space comes from a norm.

What Is a Norm?

A norm on a real or complex vector space \(V\) is a function

\[ \|\cdot\|:V\to[0,\infty) \]

satisfying three axioms for all \(x,y\in V\) and all scalars \(\alpha\).

  1. Positive definiteness: \(\|x\|\ge 0\), and \(\|x\|=0\) if and only if \(x=0\).
  2. Absolute homogeneity: \(\|\alpha x\|=|\alpha|\,\|x\|\).
  3. Triangle inequality: \(\|x+y\|\le \|x\|+\|y\|\).

Norms Measure Size

In ordinary Euclidean space,

\[ x=(x_1,\ldots,x_n) \]

has familiar length

\[ \|x\|_2= \sqrt{x_1^2+\cdots+x_n^2}. \]

The norm concept keeps the essential algebraic properties of length while allowing many different formulas for measuring size.

The Euclidean Norm

For \(x=(x_1,\ldots,x_n)\in\mathbb{R}^n\), the Euclidean norm is

\[ \boxed{ \|x\|_2= \left(\sum_{k=1}^{n}|x_k|^2\right)^{1/2} }. \]

In \(\mathbb{R}^2\),

\[ \|(x,y)\|_2=\sqrt{x^2+y^2}. \]

This is the usual distance from the origin.

The \(1\)-Norm

Another common norm on \(\mathbb{R}^n\) is

\[ \boxed{ \|x\|_1=\sum_{k=1}^{n}|x_k| }. \]

For \(x=(3,-2,5)\),

\[ \|x\|_1=3+2+5=10. \]

The \(1\)-norm adds absolute coordinate magnitudes instead of combining them through a square root.

The Infinity Norm

The infinity norm, also called the maximum norm, is

\[ \boxed{ \|x\|_\infty=\max_{1\le k\le n}|x_k| }. \]

For

\[ x=(3,-2,5), \]

we have

\[ \|x\|_\infty=5. \]

The General \(p\)-Norm

For \(1\le p<\infty\), define

\[ \boxed{ \|x\|_p= \left(\sum_{k=1}^{n}|x_k|^p\right)^{1/p} }. \]

This family includes the \(1\)-norm when \(p=1\) and the Euclidean norm when \(p=2\).

The infinity norm may be viewed as a limiting member of this family.

Example: Comparing Norms

Let

\[ x=(1,-2,2). \]

Then

\[ \|x\|_1=1+2+2=5, \] \[ \|x\|_2=\sqrt{1+4+4}=3, \]

and

\[ \|x\|_\infty=2. \]

Different norms assign different numerical sizes to the same vector, but all three satisfy the norm axioms.

A Norm Produces a Metric

Every norm defines a distance by

\[ \boxed{d(x,y)=\|x-y\|}. \]

Positive definiteness, symmetry, and the metric triangle inequality follow from the norm axioms.

In particular,

\[ d(x,y)=\|x-y\|=\|-(y-x)\|=\|y-x\|. \]

Translation Invariance

Metrics induced by norms are translation invariant.

For any \(x,y,z\in V\),

\[ d(x+z,y+z) = \|(x+z)-(y+z)\| = \|x-y\| = d(x,y). \]

Shifting both points by the same vector does not change their distance.

Convergence in a Normed Space

A sequence \((x_n)\) converges to \(x\) in norm if

\[ \boxed{\|x_n-x\|\to 0}. \]

Equivalently, for every \(\varepsilon>0\), there exists \(N\) such that

\[ n\ge N \quad\Longrightarrow\quad \|x_n-x\|<\varepsilon. \]

This is exactly metric convergence for the induced metric.

Cauchy Sequences in a Normed Space

A sequence \((x_n)\) is Cauchy if for every \(\varepsilon>0\), there exists \(N\) such that

\[ m,n\ge N \quad\Longrightarrow\quad \|x_m-x_n\|<\varepsilon. \]

The sequence eventually becomes internally close, even before a candidate limit is known.

Banach Spaces

A normed vector space is called a Banach space if it is complete with respect to the metric induced by its norm.

In other words, every Cauchy sequence in the space must converge to an element of that same space.

\[ \boxed{ \text{Banach space} = \text{complete normed vector space} }. \]

Finite-Dimensional Normed Spaces

Every finite-dimensional normed vector space over \(\mathbb{R}\) or \(\mathbb{C}\) is complete.

Therefore,

\[ \mathbb{R}^n \]

equipped with any norm is a Banach space.

This fact is special to finite dimensions. Infinite-dimensional normed spaces may fail to be complete.

Equivalent Norms in Finite Dimensions

On a finite-dimensional vector space, any two norms are equivalent in the sense that they control one another up to positive constant factors.

If \(\|\cdot\|_a\) and \(\|\cdot\|_b\) are norms on a finite-dimensional space, then there exist constants \(c,C>0\) such that

\[ c\|x\|_a \le \|x\|_b \le C\|x\|_a \]

for every \(x\).

Consequently, the norms generate the same convergent sequences and the same topology.

Norms on Function Spaces

Norms are not limited to coordinate vectors.

For continuous functions on a closed interval \([a,b]\), one common choice is the supremum norm

\[ \boxed{ \|f\|_\infty = \max_{x\in[a,b]}|f(x)| }. \]

This measures the largest magnitude attained by the function.

The Integral \(1\)-Norm

Another way to measure a function is

\[ \|f\|_1 = \int_a^b |f(x)|\,dx. \]

This norm measures total absolute area rather than maximum height.

The Integral \(2\)-Norm

A particularly important function norm is

\[ \boxed{ \|f\|_2 = \left( \int_a^b |f(x)|^2\,dx \right)^{1/2} }. \]

This norm is generated by an inner product and therefore carries a geometric structure analogous to Euclidean space.

What Is an Inner Product?

An inner product assigns a scalar to a pair of vectors.

On a real vector space \(V\), an inner product is a function

\[ \langle\cdot,\cdot\rangle:V\times V\to\mathbb{R} \]

satisfying, for all \(x,y,z\in V\) and scalars \(\alpha,\beta\):

  1. Symmetry: \(\langle x,y\rangle=\langle y,x\rangle\).
  2. Linearity: \(\langle \alpha x+\beta y,z\rangle=\alpha\langle x,z\rangle+\beta\langle y,z\rangle\).
  3. Positive definiteness: \(\langle x,x\rangle\ge0\), with equality only when \(x=0\).

Complex Inner Products

In complex vector spaces, symmetry is replaced by conjugate symmetry:

\[ \boxed{ \langle x,y\rangle = \overline{\langle y,x\rangle} }. \]

The inner product is linear in one argument and conjugate-linear in the other. Different textbooks choose which argument is linear, so the convention must be stated.

The Standard Dot Product

In \(\mathbb{R}^n\), the standard inner product is

\[ \boxed{ \langle x,y\rangle = \sum_{k=1}^{n}x_k y_k }. \]

For

\[ x=(1,2,-1), \qquad y=(3,0,4), \]

we obtain

\[ \langle x,y\rangle = 1(3)+2(0)+(-1)(4) = -1. \]

An Inner Product Produces a Norm

Every inner product defines a norm by

\[ \boxed{ \|x\| = \sqrt{\langle x,x\rangle} }. \]

For the standard inner product on \(\mathbb{R}^n\), this produces the familiar Euclidean norm.

Not Every Norm Comes from an Inner Product

Every inner-product space is a normed space, but the converse is false.

For example, on \(\mathbb{R}^2\),

\[ \|(x,y)\|_1=|x|+|y| \]

is a valid norm, but it does not arise from any inner product.

The Parallelogram Law

A norm induced by an inner product must satisfy

\[ \boxed{ \|x+y\|^2+\|x-y\|^2 = 2\|x\|^2+2\|y\|^2 }. \]

This is the parallelogram law.

A norm comes from an inner product precisely when this identity holds for all vectors in the space.

Orthogonality

Two vectors \(x\) and \(y\) are orthogonal when

\[ \boxed{\langle x,y\rangle=0}. \]

We write

\[ x\perp y. \]

Orthogonality generalizes the familiar notion of a right angle.

The Pythagorean Identity

If \(x\perp y\), then

\[ \boxed{ \|x+y\|^2 = \|x\|^2+\|y\|^2 }. \]

This follows by expanding the inner product:

\[ \|x+y\|^2 = \langle x+y,x+y\rangle. \]

The Cauchy–Schwarz Inequality

One of the fundamental inequalities of analysis is

\[ \boxed{ |\langle x,y\rangle| \le \|x\|\,\|y\| }. \]

Equality holds precisely when \(x\) and \(y\) are linearly dependent, with the usual zero-vector cases included.

Cauchy–Schwarz is what makes the inner-product norm satisfy the triangle inequality.

The Triangle Inequality from Cauchy–Schwarz

Using

\[ \|x+y\|^2 = \|x\|^2 + 2\operatorname{Re}\langle x,y\rangle + \|y\|^2, \]

Cauchy–Schwarz gives

\[ \|x+y\|^2 \le \|x\|^2+2\|x\|\,\|y\|+\|y\|^2. \]

Therefore,

\[ \boxed{ \|x+y\| \le \|x\|+\|y\| }. \]

Angles from Inner Products

For nonzero vectors in a real inner-product space, the angle \(\theta\) may be defined through

\[ \cos\theta = \frac{\langle x,y\rangle} {\|x\|\,\|y\|}. \]

Cauchy–Schwarz guarantees that the right side lies between \(-1\) and \(1\).

Orthogonal Sets

A collection of nonzero vectors

\[ \{v_1,\ldots,v_n\} \]

is orthogonal if

\[ \langle v_i,v_j\rangle=0 \qquad \text{whenever }i\ne j. \]

Orthogonal nonzero vectors are automatically linearly independent.

Orthonormal Sets

An orthogonal set is orthonormal if every vector also has norm \(1\).

\[ \langle e_i,e_j\rangle = \begin{cases} 1,& i=j,\\ 0,& i\ne j. \end{cases} \]

This compact relationship is often written

\[ \langle e_i,e_j\rangle=\delta_{ij}. \]

Coordinates in an Orthonormal Basis

If \(e_1,\ldots,e_n\) is an orthonormal basis, then every vector \(x\) can be written as

\[ \boxed{ x = \sum_{k=1}^{n} \langle x,e_k\rangle e_k }. \]

The inner products \(\langle x,e_k\rangle\) are the coordinates of \(x\) relative to that basis.

Projection onto a Single Vector

Let \(u\ne0\). The orthogonal projection of \(x\) onto the line spanned by \(u\) is

\[ \boxed{ \operatorname{proj}_u x = \frac{\langle x,u\rangle} {\langle u,u\rangle}u }. \]

If \(u\) is a unit vector, this simplifies to

\[ \operatorname{proj}_u x = \langle x,u\rangle u. \]

Projection onto a Subspace

Suppose \(M\) has orthonormal basis

\[ e_1,\ldots,e_k. \]

Then the orthogonal projection of \(x\) onto \(M\) is

\[ \boxed{ P_Mx = \sum_{j=1}^{k} \langle x,e_j\rangle e_j }. \]

The error

\[ x-P_Mx \]

is orthogonal to every vector in \(M\).

Projection as Best Approximation

Orthogonal projection is not merely geometric. It solves a minimization problem.

If \(y=P_Mx\), then

\[ \|x-y\| \le \|x-z\| \]

for every \(z\in M\).

Thus the projection is the closest point in the subspace to \(x\).

Function-Space Inner Products

For suitable functions on \([a,b]\), define

\[ \boxed{ \langle f,g\rangle = \int_a^b f(x)\overline{g(x)}\,dx }. \]

In the real-valued case, the complex conjugate may be omitted.

The associated norm is

\[ \|f\|_2 = \left( \int_a^b|f(x)|^2\,dx \right)^{1/2}. \]

Orthogonal Functions

Two functions \(f\) and \(g\) are orthogonal when

\[ \int_a^b f(x)\overline{g(x)}\,dx = 0. \]

This is the function-space version of perpendicular vectors.

Orthogonal function families are central in Fourier series, spectral decomposition, approximation theory, and differential equations.

Example: Two Orthogonal Functions

On \([-\pi,\pi]\), consider

\[ f(x)=\sin x, \qquad g(x)=\cos x. \]

Their inner product is

\[ \int_{-\pi}^{\pi} \sin x\cos x\,dx = 0. \]

Therefore,

\[ \sin x\perp \cos x. \]

Hilbert Spaces

An inner-product space that is complete in the norm induced by its inner product is called a Hilbert space.

\[ \boxed{ \text{Hilbert space} = \text{complete inner-product space} }. \]

Every Hilbert space is a Banach space, but not every Banach space is a Hilbert space.

Banach Space versus Hilbert Space

The hierarchy is

\[ \text{inner product} \Longrightarrow \text{norm} \Longrightarrow \text{metric}. \]

With completeness added,

\[ \text{Hilbert space} \Longrightarrow \text{Banach space} \Longrightarrow \text{complete metric space}. \]

The reverse implications generally fail.

Examples of Banach Spaces

Some of these spaces are Hilbert spaces for special choices of \(p\), while others are not.

Sequence Spaces \(\ell^p\)

For \(1\le p<\infty\), the sequence space \(\ell^p\) consists of sequences

\[ x=(x_1,x_2,\ldots) \]

satisfying

\[ \sum_{k=1}^{\infty}|x_k|^p<\infty. \]

The norm is

\[ \|x\|_p = \left( \sum_{k=1}^{\infty}|x_k|^p \right)^{1/p}. \]

The Space \(\ell^\infty\)

The space \(\ell^\infty\) consists of bounded sequences:

\[ \ell^\infty = \left\{ x=(x_k): \sup_k|x_k|<\infty \right\}. \]

Its norm is

\[ \boxed{ \|x\|_\infty = \sup_k|x_k| }. \]

This is a Banach space.

The Space \(\ell^2\)

Among the \(\ell^p\) spaces, \(\ell^2\) has a distinguished role because its norm comes from the inner product

\[ \langle x,y\rangle = \sum_{k=1}^{\infty} x_k\overline{y_k}. \]

Therefore, \(\ell^2\) is a Hilbert space.

The Space \(L^2\)

The function space \(L^2\) contains measurable functions whose squared magnitude is integrable:

\[ \int |f|^2<\infty. \]

With inner product

\[ \langle f,g\rangle = \int f\overline{g}, \]

the space \(L^2\) is a Hilbert space after functions that agree almost everywhere are identified.

Measure theory will make this statement precise in a later lesson.

Why Completeness Matters

Functional analysis frequently constructs an object as the limit of a sequence of approximations.

Completeness guarantees that a Cauchy process does not escape the space.

This is essential in:

Norm Convergence versus Coordinate Convergence

In finite-dimensional spaces, coordinatewise convergence and norm convergence agree.

In infinite-dimensional spaces, analogous componentwise or pointwise convergence may be weaker than norm convergence.

This distinction is one reason infinite-dimensional analysis requires careful attention to which norm or topology is being used.

Common Error: Confusing Norm and Inner Product

A norm takes one vector and returns a nonnegative real number:

\[ x\mapsto\|x\|. \]

An inner product takes two vectors and returns a scalar:

\[ (x,y)\mapsto\langle x,y\rangle. \]

An inner product can generate a norm, but the two objects are not the same.

Common Error: Assuming Every Norm Comes from an Inner Product

The \(1\)-norm and infinity norm on \(\mathbb{R}^n\) are valid norms, but for dimensions greater than one they do not satisfy the parallelogram law.

Therefore, they do not arise from inner products.

Common Error: Forgetting Completeness

A normed space is not automatically a Banach space.

A Banach space must also be complete.

Likewise, an inner-product space is not automatically a Hilbert space; it becomes Hilbert only after completeness is established.

Common Error: Treating All Function Norms as Equivalent

On infinite-dimensional spaces, different norms can produce genuinely different notions of convergence.

For example, convergence in the supremum norm is stronger than many integral forms of convergence.

Finite-dimensional norm equivalence should not be extended automatically to infinite-dimensional settings.

A Reliable Procedure for Checking a Norm

  1. Verify that \(\|x\|\ge0\).
  2. Check that \(\|x\|=0\) only for the zero vector.
  3. Verify absolute homogeneity: \(\|\alpha x\|=|\alpha|\|x\|\).
  4. Prove the triangle inequality.
  5. Only then conclude that the proposed formula is a norm.

A Reliable Procedure for Checking an Inner Product

  1. Check symmetry or conjugate symmetry.
  2. Verify linearity in the designated argument.
  3. Verify conjugate linearity in the other argument when the field is complex.
  4. Check \(\langle x,x\rangle\ge0\).
  5. Verify that \(\langle x,x\rangle=0\) implies \(x=0\).

A Reliable Procedure for Projection

  1. Identify the subspace or direction onto which the vector will be projected.
  2. Use an orthonormal basis whenever possible.
  3. Compute the coefficients \(\langle x,e_k\rangle\).
  4. Form the projection from those coefficients.
  5. Check that the error \(x-Px\) is orthogonal to the target subspace.

Practice 1: Compute Three Norms

Let

\[ x=(2,-1,2). \]

Compute \(\|x\|_1\), \(\|x\|_2\), and \(\|x\|_\infty\).

Practice 2: Induced Distance

In \(\mathbb{R}^2\) with the \(1\)-norm, find the distance between

\[ x=(1,3) \qquad\text{and}\qquad y=(-2,5). \]

Practice 3: Inner Product

Using the standard inner product on \(\mathbb{R}^3\), compute

\[ \langle(1,-2,3),(4,1,-1)\rangle. \]

Practice 4: Orthogonality

Determine whether

\[ x=(1,2) \qquad\text{and}\qquad y=(2,-1) \]

are orthogonal.

Practice 5: Projection

Find the projection of

\[ x=(3,4) \]

onto

\[ u=(1,0). \]

Practice 6: Projection onto a Nonunit Vector

Find

\[ \operatorname{proj}_u x \]

for

\[ x=(4,2), \qquad u=(1,1). \]

Practice 7: Parallelogram Law

Test the \(1\)-norm on \(\mathbb{R}^2\) using

\[ x=(1,0), \qquad y=(0,1), \]

and determine whether the parallelogram law holds.

Practice 8: Function Inner Product

On \([-1,1]\), compute

\[ \langle 1,x\rangle = \int_{-1}^{1}x\,dx. \]

Are the functions \(1\) and \(x\) orthogonal?

Practice 9: Identify the Space

Classify each description as normed space, Banach space, inner-product space, or Hilbert space when the strongest stated classification is known.

  1. A complete normed vector space.
  2. A complete inner-product space.
  3. A vector space equipped with an inner product, without a completeness assumption.

Practice 10: Sequence in \(\ell^2\)

Determine whether

\[ x= \left( 1,\frac12,\frac13,\frac14,\ldots \right) \]

belongs to \(\ell^2\).

Practice Answers

Practice 1

\[ \|x\|_1=2+1+2=5, \] \[ \|x\|_2=\sqrt{4+1+4}=3, \]

and

\[ \|x\|_\infty=2. \]

Practice 2

First compute

\[ x-y=(3,-2). \]

Therefore,

\[ d(x,y) = \|x-y\|_1 = |3|+|-2| = \boxed{5}. \]

Practice 3

\[ \langle(1,-2,3),(4,1,-1)\rangle = 4-2-3 = \boxed{-1}. \]

Practice 4

\[ \langle x,y\rangle = 1(2)+2(-1) = 0. \]

Therefore,

\[ \boxed{x\perp y}. \]

Practice 5

Since \(u=(1,0)\) is a unit vector,

\[ \operatorname{proj}_u x = \langle x,u\rangle u. \]

Now

\[ \langle(3,4),(1,0)\rangle=3, \]

so

\[ \boxed{ \operatorname{proj}_u x=(3,0) }. \]

Practice 6

\[ \langle x,u\rangle=4+2=6 \]

and

\[ \langle u,u\rangle=1+1=2. \]

Therefore,

\[ \operatorname{proj}_u x = \frac{6}{2}(1,1) = \boxed{(3,3)}. \]

Practice 7

Under the \(1\)-norm,

\[ \|x+y\|_1=2, \qquad \|x-y\|_1=2. \]

Hence

\[ \|x+y\|_1^2+\|x-y\|_1^2 = 4+4 = 8. \]

But

\[ 2\|x\|_1^2+2\|y\|_1^2 = 2+2 = 4. \]

The parallelogram law fails, so the \(1\)-norm does not come from an inner product.

Practice 8

\[ \int_{-1}^{1}x\,dx=0. \]

Therefore,

\[ \boxed{1\perp x} \]

in this inner-product space.

Practice 9

  1. A complete normed vector space is a Banach space.
  2. A complete inner-product space is a Hilbert space.
  3. Without completeness, the third is an inner-product space.

Practice 10

We test

\[ \sum_{n=1}^{\infty} \left|\frac1n\right|^2 = \sum_{n=1}^{\infty}\frac1{n^2}. \]

This \(p\)-series converges because \(p=2>1\).

Therefore,

\[ \boxed{x\in\ell^2}. \]

What to Remember

A norm measures vector size:

\[ \|x\|. \]

A norm generates a metric:

\[ d(x,y)=\|x-y\|. \]

An inner product generates a norm:

\[ \|x\|=\sqrt{\langle x,x\rangle}. \]

Completeness produces two central spaces:

\[ \boxed{ \text{Banach} = \text{complete normed space} } \]

and

\[ \boxed{ \text{Hilbert} = \text{complete inner-product space} }. \]

The key structural chain is

\[ \text{inner product} \Longrightarrow \text{norm} \Longrightarrow \text{metric}. \]

Where We Go Next

Norms and inner products provide the geometric framework needed to discuss distance, convergence, approximation, and orthogonality in spaces of functions.

The next lesson, Continuity and Compactness, develops these ideas at a more structural level by examining continuous mappings, compact sets, sequential compactness, and the powerful consequences compactness provides in analysis.