Inner Product Spaces
MAT240H1 had no way to say how long a vector is or what angle two vectors make. An inner product supplies both from a single operation. Everything geometric in this course — orthogonality, projection, the spectral theorem — is built on the axioms below.
- State the real and complex inner product axioms and check them.
- Explain why the complex case needs conjugate symmetry.
- Prove Cauchy–Schwarz and the triangle inequality.
- Work in a function space with an integral inner product.
A vector space alone cannot express "closest", "perpendicular", or "best approximation". Those are the questions applications actually ask, and an inner product is the minimum extra structure that answers them.
Least squares regression minimises a distance defined by an inner product. Fourier series are orthogonal expansions in a function space. Quantum states live in a complex inner product space, where $\langle u,v\rangle$ is a probability amplitude — which is why the complex case matters.
Cauchy proved the finite sum version in 1821; Bunyakovsky and Schwarz found the integral version decades later, which is why the inequality carries up to three names. Extending these axioms to infinite dimensions and demanding completeness gives Hilbert space, the subject of MAT357H1.
The axioms
Let $V$ be a vector space over $F=\mathbb{R}$ or $\mathbb{C}$. An inner product is a map $\langle\cdot,\cdot\rangle:V\times V\to F$ with, for all $u,v,w\in V$ and $c\in F$:
The norm is $\|u\|=\sqrt{\langle u,u\rangle}$.
$\langle u,cv\rangle=\overline{c}\,\langle u,v\rangle$ and $\langle u,v+w\rangle=\langle u,v\rangle+\langle u,w\rangle$.
This is forced, not chosen: conjugate symmetry plus linearity in the first slot gives $\langle u,cv\rangle=\overline{\langle cv,u\rangle} =\overline{c\langle v,u\rangle}=\overline{c}\,\overline{\langle v,u\rangle} =\overline{c}\,\langle u,v\rangle$.
The standard examples
Cauchy–Schwarz and its consequences
$|\langle u,v\rangle|\le\|u\|\,\|v\|$, with equality exactly when $u$ and $v$ are linearly dependent.
$\|u+v\|\le\|u\|+\|v\|$.
Expand $\|u+v\|^2=\|u\|^2+2\operatorname{Re}\langle u,v\rangle+\|v\|^2$. Since $\operatorname{Re}\langle u,v\rangle\le|\langle u,v\rangle| \le\|u\|\|v\|$ by Cauchy–Schwarz, the right side is at most $(\|u\|+\|v\|)^2$.
$\|u+v\|^2+\|u-v\|^2=2\|u\|^2+2\|v\|^2$. Expanding both squares makes the cross terms cancel. A norm satisfies this identity if and only if it comes from an inner product, which is how you rule out $\|\cdot\|_1$.
• Pulling a scalar out of the second slot without conjugating. Over
$\mathbb{C}$, $\langle u,cv\rangle=\overline{c}\langle u,v\rangle$.
• Assuming $\langle u,v\rangle$ is real. It is a complex number; only
$\langle u,u\rangle$ is guaranteed real.
• Defining the angle between complex vectors. The formula
$\cos\theta=\langle u,v\rangle/(\|u\|\|v\|)$ needs a real inner product;
over $\mathbb{C}$ only orthogonality survives.
• Forgetting that the inner product is extra data. The same vector space
carries many different inner products, and "orthogonal" means nothing until you
name one.
Practice
Original problems written for this guide in the style of Friedberg, Insel & Spence exercises. Write a full solution before opening the answer.
On $\mathbb{R}^2$, is $\langle u,v\rangle=u_1v_1-u_2v_2$ an inner product?
Answer
For which $a,b\in\mathbb{R}$ is $\langle u,v\rangle=au_1v_1+bu_2v_2$ an inner product on $\mathbb{R}^2$?
Answer
In $\mathbb{C}^2$ with the standard inner product, compute $\langle u,v\rangle$ and $\langle v,u\rangle$ for $u=(1,i)$, $v=(2,1-i)$.
Solution
In $C[0,1]$ with $\langle f,g\rangle=\int_0^1 fg$, compute $\|f\|$ for $f(t)=t$.
Solution
In a real inner product space, prove $\langle u,v\rangle=\tfrac14\left(\|u+v\|^2-\|u-v\|^2\right)$.
Solution
Show $\|u\|_1=|u_1|+|u_2|$ on $\mathbb{R}^2$ comes from no inner product.
Solution
When is $\|u+v\|=\|u\|+\|v\|$ in a real inner product space?
Solution
Verify $\langle A,B\rangle=\operatorname{tr}(B^*A)$ is positive definite on $M_n(\mathbb{C})$.
Solution
Expanded topic checklist
Course coverage. This unit is part of the complete MAT247H1 topic map. It should cover the following items before you move on:
- Axioms. State linearity, conjugate symmetry and positive definiteness, and check them on a proposed form.
- Real versus complex. Explain why conjugation is forced in the complex case and where the second slot becomes conjugate-linear.
- Cauchy–Schwarz. Reproduce the proof and identify the equality case.
- Norms from inner products. Derive the triangle inequality and the parallelogram law, and use the latter as a test.
Use the current UofT MAT247H1 calendar topics as the coverage spine, and pair lecture notes with a Friedberg-Insel-Spence-style linear algebra text for extra reading. The exact assigned textbook should still be checked against the instructor syllabus.
The spine here is short: the axioms give Cauchy–Schwarz, which gives the triangle inequality, which makes $\|\cdot\|$ a genuine norm and $V$ a metric space.
Deep practice extensions
Use these after the main practice set. They are designed to force definition-level reasoning, theorem selection, and mixed application rather than routine calculation only.
If $\langle cu,v\rangle=c\langle u,v\rangle$, what is $\langle u,cv\rangle$?
Guide
Which axiom fails for $\langle u,v\rangle=u_1v_1$ on $\mathbb{R}^2$?
Guide
State when Cauchy–Schwarz is an equality.
Guide
How do you decide whether a given norm arises from an inner product?