The Central Idea
A function space gives us a collection of vectors. An operator tells us how those vectors are transformed.
The central progression is
\[ \boxed{ \text{vector spaces} \longrightarrow \text{linear operators} \longrightarrow \text{bounded operators} \longrightarrow \text{linear functionals and duality} }. \]Functional analysis asks which operators preserve enough structure to support stable limiting arguments.
What Is an Operator?
An operator is a map between spaces.
If \(X\) and \(Y\) are vector spaces, an operator may be written
\[ T:X\to Y. \]When \(X\) and \(Y\) are function spaces, the input and output may themselves be functions.
Linear Operators
An operator
\[ T:X\to Y \]is linear if for all \(x,y\in X\) and scalars \(a,b\),
\[ \boxed{ T(ax+by)=aT(x)+bT(y) }. \]Equivalently, linearity combines additivity and homogeneity.
Additivity and Homogeneity
Additivity means
\[ T(x+y)=T(x)+T(y), \]while homogeneity means
\[ T(cx)=cT(x). \]Together they imply the full linearity rule
\[ T(ax+by)=aT(x)+bT(y). \]Immediate Consequences of Linearity
Every linear operator satisfies
\[ T(0)=0. \]Also,
\[ T(-x)=-T(x), \]and for any finite linear combination,
\[ T\left( \sum_{k=1}^{n}a_kx_k \right) = \sum_{k=1}^{n}a_kT(x_k). \]Example: Multiplication Operator
Fix a function \(m\). Define
\[ (Tf)(x)=m(x)f(x). \]Then
\[ T(af+bg) = m(af+bg) = aTf+bTg. \]Therefore multiplication by a fixed function is linear.
Example: Integration as an Operator
Define
\[ (Tf)(x) = \int_a^x f(t)\,dt. \]Then
\[ T(af+bg)(x) = a\int_a^x f(t)\,dt + b\int_a^x g(t)\,dt. \]Hence \(T\) is linear.
Example: Differentiation as a Linear Operator
On a suitable differentiable function space, define
\[ Df=f'. \]Then
\[ D(af+bg) = af'+bg' = aDf+bDg. \]Therefore differentiation is linear.
The Domain Matters
An operator is not determined only by a formula. Its domain and codomain are part of its definition.
For example,
\[ Df=f' \]makes sense as a map
\[ D:C^1([a,b])\to C([a,b]), \]but not as a map from all of \(C([a,b])\), because not every continuous function is differentiable.
Kernel of a Linear Operator
The kernel of
\[ T:X\to Y \]is
\[ \boxed{ \ker T = \{x\in X:T(x)=0\} }. \]The kernel is always a linear subspace of \(X\).
Range of a Linear Operator
The range is
\[ \boxed{ \operatorname{ran}T = \{T(x):x\in X\} }. \]The range is a linear subspace of \(Y\).
Injectivity and the Kernel
A linear operator \(T\) is injective exactly when
\[ \boxed{ \ker T=\{0\} }. \]Thus injectivity can be tested by solving the homogeneous equation
\[ T(x)=0. \]Example: Kernel of Differentiation
Let
\[ D:C^1([a,b])\to C([a,b]) \]be differentiation.
Then
\[ Df=0 \]exactly when \(f\) is constant.
Therefore,
\[ \boxed{ \ker D = \{\text{constant functions}\} }. \]Bounded Linear Operators
Let \(X\) and \(Y\) be normed spaces. A linear operator
\[ T:X\to Y \]is bounded if there exists \(C\ge0\) such that
\[ \boxed{ \|Tx\|_Y \le C\|x\|_X } \]for every \(x\in X\).
Bounded Does Not Mean the Range Is Bounded
The word bounded refers to controlled growth relative to the input norm.
It does not mean that
\[ \{Tx:x\in X\} \]is a bounded set.
If \(T\ne0\), scalar multiplication of the input typically makes \(\|Tx\|\) arbitrarily large.
Boundedness and Continuity
For linear maps between normed spaces,
\[ \boxed{ T\text{ is bounded} \iff T\text{ is continuous} }. \]This equivalence is one of the fundamental simplifications of functional analysis.
Why Boundedness Implies Continuity
If
\[ \|Tx\|\le C\|x\|, \]then
\[ \|Tx-Ty\| = \|T(x-y)\| \le C\|x-y\|. \]Thus every bounded linear operator is Lipschitz continuous.
Continuity at One Point Is Enough
A linear operator is continuous everywhere if it is continuous at even one point.
In particular, continuity at \(0\) is sufficient.
Linearity transfers local control at the origin throughout the entire space.
The Operator Norm
For a bounded linear operator \(T:X\to Y\), define
\[ \boxed{ \|T\| = \sup_{\|x\|\le1}\|Tx\| }. \]Equivalently,
\[ \|T\| = \sup_{\|x\|=1}\|Tx\| \]when \(X\ne\{0\}\).
The Operator Norm as the Best Bound
The operator norm is the smallest constant \(C\) satisfying
\[ \|Tx\| \le C\|x\| \]for every \(x\in X\).
Thus
\[ \boxed{ \|Tx\| \le \|T\|\|x\| }. \]Example: Scalar Multiplication
Let
\[ T(x)=cx \]on a normed space.
Then
\[ \|Tx\| = |c|\|x\|. \]Therefore,
\[ \boxed{\|T\|=|c|}. \]Example: Evaluation on \(C([a,b])\)
Fix \(x_0\in[a,b]\). Define
\[ E_{x_0}(f)=f(x_0). \]Then
\[ |E_{x_0}(f)| = |f(x_0)| \le \|f\|_\infty. \]Hence \(E_{x_0}\) is bounded and
\[ \boxed{ \|E_{x_0}\|=1 }. \]Linear Functionals
A linear functional is a linear map from a vector space into its scalar field.
Over the real numbers,
\[ \boxed{ \varphi:X\to\mathbb{R} }. \]Over the complex numbers, the codomain is \(\mathbb{C}\).
Example: Integral Functional
On \(C([a,b])\), define
\[ \varphi(f) = \int_a^b f(x)\,dx. \]Then
\[ |\varphi(f)| \le \int_a^b|f(x)|\,dx \le (b-a)\|f\|_\infty. \]Therefore \(\varphi\) is bounded and
\[ \|\varphi\| \le b-a. \]The Exact Norm of the Integral Functional
For the functional
\[ \varphi(f)=\int_a^b f(x)\,dx \]on \(C([a,b])\) with the supremum norm, choose \(f\equiv1\).
Then
\[ \|f\|_\infty=1 \]and
\[ |\varphi(f)|=b-a. \]Hence
\[ \boxed{ \|\varphi\|=b-a }. \]Coordinate Functionals on Sequence Spaces
On \(\ell^p\), define
\[ \pi_k(x)=x_k. \]Then
\[ |x_k| \le \|x\|_p \]for \(1\le p\le\infty\).
Therefore each coordinate map is a bounded linear functional.
The Dual Space
Let \(X\) be a normed space. The collection of all bounded linear functionals on \(X\) is called the dual space and is denoted
\[ \boxed{X^*}. \]The dual space is itself a normed vector space under
\[ \|\varphi\| = \sup_{\|x\|\le1}|\varphi(x)|. \]The Dual Space Is Always Banach
Even when \(X\) is not complete, its dual space \(X^*\) is complete in the operator norm.
Therefore,
\[ \boxed{ X^* \text{ is always a Banach space} }. \]Dual Pairing
If \(\varphi\in X^*\) and \(x\in X\), the scalar
\[ \varphi(x) \]is often written as a pairing:
\[ \langle \varphi,x\rangle. \]This notation emphasizes the interaction between a space and its dual.
Duality in \(\ell^p\)
For
\[ 1every \(y\in\ell^q\) defines a bounded functional on \(\ell^p\) by
\[ \varphi_y(x) = \sum_{n=1}^{\infty}x_ny_n. \]Hölder's inequality gives
\[ |\varphi_y(x)| \le \|x\|_p\|y\|_q. \]The Dual of \(\ell^p\)
For
\[ 1where \(1/p+1/q=1\).
Also,
\[ \boxed{ (\ell^1)^* \cong \ell^\infty }. \]Duality in \(L^p\)
Under standard hypotheses on the measure space, if
\[ 1Hölder's inequality yields
\[ |\varphi_g(f)| \le \|f\|_p\|g\|_q. \]Riesz Representation in Hilbert Spaces
Let \(H\) be a Hilbert space. Every bounded linear functional
\[ \varphi:H\to\mathbb{R} \]can be represented using the inner product.
There exists a unique \(y\in H\) such that
\[ \boxed{ \varphi(x)=\langle x,y\rangle } \]for every \(x\in H\).
Norm in the Riesz Representation Theorem
If
\[ \varphi(x)=\langle x,y\rangle, \]then
\[ \boxed{ \|\varphi\|=\|y\| }. \]Thus a Hilbert space and its dual are naturally linked through the inner product.
The Hahn-Banach Principle
One of the foundational results of functional analysis is the Hahn-Banach theorem.
In one standard form, it says that a bounded linear functional defined on a subspace can be extended to the entire normed space without increasing its norm.
Symbolically, if
\[ \varphi:Y\to\mathbb{R} \]is bounded and linear on a subspace \(Y\subseteq X\), then there exists an extension
\[ \Phi:X\to\mathbb{R} \]satisfying
\[ \Phi|_Y=\varphi \]and
\[ \boxed{ \|\Phi\|=\|\varphi\| }. \]Why Hahn-Banach Matters
Hahn-Banach ensures that sufficiently many bounded linear functionals exist to distinguish vectors and probe the geometry of normed spaces.
A major consequence is that for every nonzero \(x\in X\), there exists \(\varphi\in X^*\) such that
\[ \varphi(x)\ne0. \]The dual space therefore carries meaningful information about the original space.
Spaces of Bounded Operators
The set of bounded linear operators from \(X\) to \(Y\) is denoted
\[ \boxed{ \mathcal{B}(X,Y) } \]or sometimes
\[ \mathcal{L}(X,Y). \]Equipped with the operator norm, this is a normed vector space.
Completeness of Operator Spaces
If \(Y\) is Banach, then
\[ \boxed{ \mathcal{B}(X,Y) \text{ is Banach} }. \]The domain \(X\) need not itself be complete.
Composition of Bounded Operators
If
\[ T:X\to Y \]and
\[ S:Y\to Z \]are bounded linear operators, then
\[ S\circ T:X\to Z \]is bounded and linear.
Moreover,
\[ \boxed{ \|S\circ T\| \le \|S\|\,\|T\| }. \]The Identity Operator
The identity operator on \(X\) is
\[ I_X(x)=x. \]It is bounded and
\[ \boxed{ \|I_X\|=1 } \]when \(X\ne\{0\}\).
Invertible Operators
A linear operator
\[ T:X\to Y \]is invertible if it is bijective and has an inverse map
\[ T^{-1}:Y\to X. \]In Banach spaces, an important theorem says that a bounded bijective linear operator has a bounded inverse.
The Bounded Inverse Theorem
If \(X\) and \(Y\) are Banach spaces and
\[ T:X\to Y \]is bounded, linear, and bijective, then
\[ \boxed{ T^{-1}:Y\to X \text{ is bounded} }. \]Algebraic invertibility therefore becomes stable analytic invertibility in the Banach-space setting.
The Open Mapping Theorem
If \(X\) and \(Y\) are Banach spaces and
\[ T:X\to Y \]is bounded, linear, and surjective, then \(T\) maps open sets to open sets.
This result is closely related to the bounded inverse theorem.
The Closed Graph Theorem
Let \(X\) and \(Y\) be Banach spaces. Suppose a linear operator
\[ T:X\to Y \]has a closed graph:
\[ x_n\to x, \qquad Tx_n\to y \]implies
\[ y=Tx. \]Then
\[ \boxed{ T\text{ is bounded} }. \]The Uniform Boundedness Principle
Let \(X\) be Banach and \(Y\) normed. Suppose
\[ \{T_\alpha\} \subseteq \mathcal{B}(X,Y) \]is a family such that for every fixed \(x\in X\),
\[ \sup_\alpha\|T_\alpha x\|<\infty. \]Then
\[ \boxed{ \sup_\alpha\|T_\alpha\|<\infty }. \]Pointwise boundedness of the whole family upgrades to uniform operator-norm boundedness.
Why Banach-Space Completeness Reappears
The bounded inverse theorem, open mapping theorem, closed graph theorem, and uniform boundedness principle all rely essentially on completeness.
These results show that Banach spaces are not merely complete metric spaces; completeness drives strong structural consequences for linear operators.
Unbounded Linear Operators
Not every important linear operator is bounded.
Differentiation is a standard example. Consider
\[ D:C^1([0,1])\to C([0,1]). \]If the domain is given only the supremum norm
\[ \|f\|_\infty, \]differentiation is not bounded with respect to that norm.
Why Differentiation Can Be Unbounded
Consider
\[ f_n(x)=\frac{\sin(nx)}{n}. \]Then
\[ \|f_n\|_\infty\le\frac1n\to0, \]while
\[ Df_n(x)=\cos(nx) \]satisfies
\[ \|Df_n\|_\infty=1. \]Thus no constant \(C\) can satisfy
\[ \|Df\|_\infty \le C\|f\|_\infty \]for all differentiable \(f\).
The Norm Can Change Boundedness
On \(C^1([0,1])\), use the norm
\[ \|f\|_{C^1} = \|f\|_\infty+\|f'\|_\infty. \]Then
\[ \|Df\|_\infty = \|f'\|_\infty \le \|f\|_{C^1}. \]Therefore
\[ D:C^1([0,1])\to C([0,1]) \]is bounded under the \(C^1\) norm.
Integral Operators
Let \(K(x,t)\) be a fixed kernel. An integral operator may take the form
\[ (Tf)(x) = \int_a^b K(x,t)f(t)\,dt. \]Such operators occur in integral equations, differential equations, signal processing, probability, and spectral theory.
A Simple Integral-Operator Bound
Suppose
\[ |K(x,t)|\le M \]for all \(x,t\in[a,b]\). Then for \(f\in C([a,b])\),
\[ |Tf(x)| \le \int_a^b M|f(t)|\,dt \le M(b-a)\|f\|_\infty. \]Hence
\[ \boxed{ \|T\| \le M(b-a) }. \]Projection Operators
In an inner-product space, projection onto a closed subspace provides a fundamental operator.
If \(M\) is a closed subspace of a Hilbert space \(H\), then each \(x\in H\) can be written
\[ x=Px+(x-Px) \]where
\[ Px\in M \]and
\[ x-Px\perp M. \]Properties of Orthogonal Projection
An orthogonal projection \(P\) satisfies
\[ P^2=P. \]It is linear and bounded, and when \(M\ne\{0\}\),
\[ \|P\|=1. \]Projection is the operator form of best approximation in Hilbert spaces.
Adjoint Operators
Let \(H\) and \(K\) be Hilbert spaces and
\[ T:H\to K \]bounded and linear.
The adjoint
\[ T^*:K\to H \]is defined by
\[ \boxed{ \langle Tx,y\rangle_K = \langle x,T^*y\rangle_H }. \]Adjoint Identities
For bounded operators on Hilbert spaces,
\[ (S+T)^*=S^*+T^*, \] \[ (cT)^*=\overline{c}\,T^*, \] \[ (ST)^*=T^*S^*, \]and
\[ \boxed{ \|T^*\|=\|T\| }. \]Self-Adjoint Operators
An operator on a Hilbert space is self-adjoint if
\[ \boxed{T=T^*}. \]Self-adjoint operators generalize real symmetric matrices and play a central role in spectral theory, quantum mechanics, and variational analysis.
Compact Operators
A bounded operator
\[ T:X\to Y \]is compact if it maps bounded sets into relatively compact sets.
Equivalently, every bounded sequence \((x_n)\) has a subsequence \((x_{n_k})\) such that
\[ Tx_{n_k} \]converges in \(Y\).
Finite-Rank Operators
An operator has finite rank if
\[ \dim(\operatorname{ran}T)<\infty. \]Every bounded finite-rank operator is compact.
Compact operators can often be understood as infinite-dimensional analogues of operators that are well approximated by finite-dimensional behavior.
Functionals as Measurements
A linear functional extracts one scalar quantity from a vector or function.
Examples include:
- evaluation at a point;
- integration against a fixed function;
- a Fourier coefficient;
- a coordinate of a sequence;
- a weighted average;
- a boundary value.
In this sense, functionals act as linear measurements.
Functionals and Weak Convergence
A sequence \(x_n\) in a normed space \(X\) converges weakly to \(x\) if
\[ \boxed{ \varphi(x_n)\to\varphi(x) } \]for every \(\varphi\in X^*\).
Weak convergence tests vectors through all bounded linear functionals rather than requiring norm convergence.
Norm Convergence Implies Weak Convergence
If
\[ \|x_n-x\|\to0, \]then for every \(\varphi\in X^*\),
\[ |\varphi(x_n)-\varphi(x)| = |\varphi(x_n-x)| \le \|\varphi\|\|x_n-x\|\to0. \]Therefore,
\[ \boxed{ \text{norm convergence} \Longrightarrow \text{weak convergence} }. \]The Converse Need Not Hold
In infinite-dimensional spaces, weak convergence is generally weaker than norm convergence.
A sequence can be detected as convergent by every bounded linear functional while remaining a fixed positive distance from its weak limit in norm.
Differential Operators and Generalized Functions
Differential operators may be unbounded on natural normed spaces, yet they remain central to analysis.
One strategy is to change the space on which the operator acts. Another is to reinterpret differentiation through its action on carefully chosen test functions.
This second idea leads directly to distributions and generalized functions.
Integration by Parts as an Operator Identity
For sufficiently regular functions with appropriate boundary behavior,
\[ \int_a^b f'(x)\phi(x)\,dx = -\int_a^b f(x)\phi'(x)\,dx. \]This suggests that even when a classical derivative of \(f\) does not exist, we may define a generalized derivative by transferring differentiation onto the test function \(\phi\).
Common Error: Linearity Does Not Imply Boundedness
In finite-dimensional normed spaces, every linear map is bounded.
In infinite-dimensional spaces, linear operators can be unbounded.
Differentiation with an unsuitable domain norm provides a standard example.
Common Error: Bounded Operator Does Not Mean Bounded Range
The inequality
\[ \|Tx\|\le C\|x\| \]controls output size relative to input size.
It does not imply that the entire range of \(T\) is a bounded subset.
Common Error: Forgetting the Domain and Codomain
The same formula can define different operators depending on the spaces and norms involved.
Boundedness, continuity, surjectivity, and even whether the formula is defined can change when the domain or codomain changes.
Common Error: Confusing Kernel with Range
The kernel consists of inputs sent to zero:
\[ \ker T=\{x:T(x)=0\}. \]The range consists of outputs actually produced:
\[ \operatorname{ran}T=\{Tx:x\in X\}. \]Common Error: Using a Bound That Is Not Uniform
To prove boundedness, one needs one constant \(C\) satisfying
\[ \|Tx\|\le C\|x\| \]for every \(x\).
A constant that depends on \(x\) does not establish boundedness.
Common Error: Assuming Every Functional Is Continuous
In finite-dimensional spaces every linear functional is continuous.
In general infinite-dimensional normed spaces, discontinuous linear functionals can exist.
The dual space \(X^*\) contains only the bounded, equivalently continuous, linear functionals.
Common Error: Confusing Weak and Norm Convergence
Weak convergence means
\[ \varphi(x_n)\to\varphi(x) \]for every bounded linear functional.
Norm convergence means
\[ \|x_n-x\|\to0. \]Norm convergence implies weak convergence, but the converse may fail.
A Reliable Procedure for Checking Linearity
- Take arbitrary \(x,y\) in the domain.
- Take arbitrary scalars \(a,b\).
- Compute \(T(ax+by)\).
- Simplify using the operator's definition.
- Verify that the result equals \(aT(x)+bT(y)\).
A Reliable Procedure for Proving Boundedness
- Write the norm of the output, \(\|Tx\|_Y\).
- Estimate it using inequalities appropriate to the spaces.
- Factor the estimate into a constant times \(\|x\|_X\).
- Make sure the constant does not depend on \(x\).
- Conclude boundedness.
- Optimize the constant when the exact operator norm is required.
A Reliable Procedure for Finding an Operator Norm
- Prove an upper bound \[ \|Tx\|\le C\|x\|. \]
- Conclude \(\|T\|\le C\).
- Search for vectors with \(\|x\|=1\) that make \(\|Tx\|\) approach or attain \(C\).
- Use those vectors to prove \(\|T\|\ge C\).
- Combine the two inequalities.
A Reliable Procedure for Analyzing a Linear Functional
- Verify linearity.
- Estimate \(|\varphi(x)|\) by a constant times \(\|x\|\).
- Conclude boundedness and membership in \(X^*\).
- Determine the smallest possible bound when the norm is required.
- Identify whether the functional has a representation through an inner product or integral pairing.
Example: Operator Norm of Evaluation
Let
\[ E_0:C([-1,1])\to\mathbb{R} \]be
\[ E_0(f)=f(0). \]Since
\[ |f(0)|\le\|f\|_\infty, \]we have
\[ \|E_0\|\le1. \]Taking \(f\equiv1\) gives
\[ |E_0(f)|=1 \]with \(\|f\|_\infty=1\). Therefore,
\[ \boxed{\|E_0\|=1}. \]Example: Integral Functional on \(L^2(0,1)\)
Define
\[ \varphi(f)=\int_0^1f(x)\,dx. \]By Cauchy-Schwarz,
\[ |\varphi(f)| = \left| \int_0^1f(x)\cdot1\,dx \right| \le \|f\|_2\|1\|_2. \]Since
\[ \|1\|_2=1, \]we obtain
\[ |\varphi(f)| \le \|f\|_2. \]Equality occurs for \(f\equiv1\), so
\[ \boxed{\|\varphi\|=1}. \]Example: A Bounded Multiplication Operator
Let \(m\in L^\infty(X,\mu)\) and define
\[ (Tf)(x)=m(x)f(x) \]on \(L^2(X,\mu)\).
Then
\[ |Tf| \le \|m\|_\infty|f| \]almost everywhere, so
\[ \|Tf\|_2 \le \|m\|_\infty\|f\|_2. \]Therefore,
\[ \boxed{ \|T\|\le\|m\|_\infty }. \]Practice 1: Linearity
Define
\[ T:\mathbb{R}^2\to\mathbb{R}^2 \]by
\[ T(x,y)=(2x-y,x+3y). \]Verify that \(T\) is linear.
Practice 2: Kernel
For the operator in Practice 1, find
\[ \ker T. \]Practice 3: Evaluation Functional
Let
\[ \varphi(f)=f(2) \]on \(C([0,3])\) with the supremum norm. Find \(\|\varphi\|\).
Practice 4: Integral Functional
Define
\[ \varphi(f) = \int_0^2f(x)\,dx \]on \(C([0,2])\) with the supremum norm. Find \(\|\varphi\|\).
Practice 5: Multiplication Operator
On \(C([0,1])\), define
\[ (Tf)(x)=x^2f(x). \]Find the operator norm of \(T\) with respect to the supremum norm.
Practice 6: Composition Bound
Suppose \(T:X\to Y\) and \(S:Y\to Z\) are bounded linear operators. Prove
\[ \|ST\| \le \|S\|\|T\|. \]Practice 7: Differentiation and Norm Choice
Explain why
\[ D:C^1([0,1])\to C([0,1]) \]is bounded if \(C^1([0,1])\) is given the norm
\[ \|f\|_{C^1} = \|f\|_\infty+\|f'\|_\infty. \]Practice 8: Dual Pairing
Let \(y=(1/2^n)\). Define
\[ \varphi_y(x) = \sum_{n=1}^{\infty}x_ny_n \]on \(\ell^2\). Explain why \(\varphi_y\) is bounded.
Practice 9: Weak Convergence
Suppose \(x_n\to x\) in norm in a normed space \(X\). Prove that
\[ x_n\rightharpoonup x \]weakly.
Practice 10: Identify the Error
A student says:
Differentiation is linear, so it must be bounded on every normed function space where it is defined.
Explain the error.
Practice Answers
Practice 1
Let \(u=(x_1,y_1)\), \(v=(x_2,y_2)\), and scalars \(a,b\). Then
\[ T(au+bv) \]distributes componentwise to
\[ aT(u)+bT(v). \]Therefore,
\[ \boxed{T\text{ is linear}}. \]Practice 2
Solve
\[ 2x-y=0 \]and
\[ x+3y=0. \]From the first equation, \(y=2x\). Substitution gives
\[ x+6x=7x=0. \]Hence \(x=0\) and \(y=0\). Therefore,
\[ \boxed{\ker T=\{(0,0)\}}. \]Practice 3
We have
\[ |\varphi(f)| = |f(2)| \le \|f\|_\infty. \]Thus \(\|\varphi\|\le1\). Taking \(f\equiv1\) gives equality, so
\[ \boxed{\|\varphi\|=1}. \]Practice 4
We have
\[ |\varphi(f)| \le \int_0^2|f(x)|\,dx \le 2\|f\|_\infty. \]Thus \(\|\varphi\|\le2\). Taking \(f\equiv1\) gives equality, so
\[ \boxed{\|\varphi\|=2}. \]Practice 5
Since \(0\le x^2\le1\) on \([0,1]\),
\[ \|Tf\|_\infty = \sup_{x\in[0,1]}|x^2f(x)| \le \|f\|_\infty. \]Thus \(\|T\|\le1\). Taking \(f\equiv1\) gives
\[ \|Tf\|_\infty=1. \]Therefore,
\[ \boxed{\|T\|=1}. \]Practice 6
For every \(x\),
\[ \|STx\| \le \|S\|\|Tx\| \le \|S\|\|T\|\|x\|. \]Therefore,
\[ \boxed{ \|ST\| \le \|S\|\|T\| }. \]Practice 7
We have
\[ \|Df\|_\infty = \|f'\|_\infty \le \|f\|_\infty+\|f'\|_\infty = \|f\|_{C^1}. \]Therefore \(D\) is bounded, with
\[ \|D\|\le1. \]Practice 8
The sequence
\[ y_n=\frac1{2^n} \]belongs to \(\ell^2\) because
\[ \sum_{n=1}^{\infty}\frac1{4^n}<\infty. \]By Cauchy-Schwarz,
\[ |\varphi_y(x)| \le \|x\|_2\|y\|_2. \]Hence \(\varphi_y\) is bounded.
Practice 9
If \(x_n\to x\) in norm, then for every \(\varphi\in X^*\),
\[ |\varphi(x_n)-\varphi(x)| = |\varphi(x_n-x)| \le \|\varphi\|\|x_n-x\|. \]The right side tends to zero, so
\[ \boxed{ \varphi(x_n)\to\varphi(x) } \]for every bounded linear functional. Hence \(x_n\rightharpoonup x\).
Practice 10
Linearity alone does not imply boundedness in infinite-dimensional spaces. Boundedness depends on the domain, codomain, and their norms.
For example, differentiation is not bounded from \(C^1([0,1])\) equipped only with the supremum norm into \(C([0,1])\), even though it is linear.
What to Remember
A linear operator satisfies
\[ \boxed{ T(ax+by)=aT(x)+bT(y) }. \]A bounded linear operator satisfies
\[ \boxed{ \|Tx\| \le C\|x\| }. \]For linear maps between normed spaces,
\[ \boxed{ \text{bounded} \iff \text{continuous} }. \]The operator norm is
\[ \boxed{ \|T\| = \sup_{\|x\|\le1}\|Tx\| }. \]A bounded linear functional belongs to the dual space
\[ \boxed{X^*}. \]Hilbert-space functionals can be represented by inner products through the Riesz representation theorem, while Hahn-Banach explains why bounded functionals can be extended and used to probe the geometry of normed spaces.
The governing habit is: always specify the domain, codomain, and norms before deciding whether an operator is bounded or continuous.
Where We Go Next
Ordinary functions and bounded operators already provide a powerful analytical language, but differential equations, impulses, point sources, discontinuous data, and limiting processes frequently produce objects that are not classical functions.
The next lesson, Distributions and Generalized Functions, enlarges the function concept by defining objects through their action on test functions. This makes the Dirac delta, weak derivatives, and many singular constructions mathematically precise.