Projecting in infinite dimensions

Mathematics · Calculus & Analysis · ages 23-24

Name ______________________ Date ____________

In R^2 with the usual dot product, project the vector (3, 4) onto the direction (1, 0). What is the first component of the projection?
Answer: ______________
What does completeness require of an inner product space?
- Every subspace is automatically closed
- Every sequence that ought to converge lands inside the space
- Every orthonormal set is finite
In a Hilbert space, Fourier series converge to functions inside the space.
Circle one: True False
You subtract your candidate projection from the vector. What must the leftover satisfy?
- It must equal the original vector
- It must lie inside the subspace
- It must be orthogonal to everything in the subspace
In R^3, a vector v has components 1, 2, 2 in an orthonormal basis. By Parseval's identity, the squared norm of v is the sum of the squared components. What is it?
Answer: ______________
Why must the subspace be closed for the closest point guarantee?
- Finite sums otherwise violate Parseval
- Limits of approximating vectors can otherwise fall outside it
- Bounded measurements otherwise become unbounded
A vector has orthonormal coefficients 6 and 8. What is its squared length?
Answer: ______________
The vector (1, 2) in R^2 is projected onto the x-axis. Which vector is the residual?
- (0, 2)
- (1, 0)
- (0, 1)
- (2, 0)
Leo drops completeness but still claims every Fourier series lands inside. What is wrong?
- He used too large a basis for the expansion
- He tested the residual against too few vectors
- He kept the wrong hypothesis: completeness is what keeps limits inside
Mia says: in a Hilbert space, every continuous linear functional equals the inner product with one fixed vector. Is Mia right?
Circle one: True False

Answer key

For grown-ups. Fold this page away before handing over the rest.

Projecting in infinite dimensions W1-mt_xmKq8SAgvy-s1

3 · The projection formula scales (1, 0) by the ratio 3/1, giving (3, 0), whose first component is 3.
Every sequence that ought to converge lands inside the space · Completeness is about limits staying inside the space.
True · Completeness is exactly what keeps those limits inside.
It must be orthogonal to everything in the subspace · The residual test is orthogonality against everything kept.
9 · Parseval gives 1 + 4 + 4 = 9, so the squared norm is 9.
Limits of approximating vectors can otherwise fall outside it · Closed means the set already contains its limit points.
100 · Parseval adds the squared sizes: 36 plus 64 is 100.
(0, 2) · The projection onto the x-axis is (1, 0), so the residual is (1, 2) minus (1, 0) = (0, 2).
He kept the wrong hypothesis: completeness is what keeps limits inside · Without completeness the limit can escape the space.
True · Mia is right: that is exactly the Riesz representation theorem, which pairs each continuous linear functional with a unique representing vector.