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Recipe

Test whether a norm comes from an inner product

Use when: you have two specific functions f,g and a candidate norm, and need to know whether that norm could ever support orthogonal projection

Goal: a verified answer, backed by one concrete check, on whether an inner product could possibly sit behind this norm

Taught in Why L2 is the right space for Fourier series

Quick recipe

  1. Compute each function's own norm, squared
  2. Compute the norm of the sum and the difference, squared
  3. Plug all four numbers into the identity and compare both sides
  4. Interpret the result, respecting the asymmetry between failing and holding
  1. 1

    For f and g, compute \lVert f\rVert^2 and \lVert g\rVert^2 under the candidate norm.

    Example

    Let f(t)=t and g(t)=1 on [-1,1], tested under \lVert\cdot\rVert_2. Then \lVert f\rVert_2^2=\displaystyle\int_{-1}^1t^2\,dt=\frac23 and \lVert g\rVert_2^2=\displaystyle\int_{-1}^1 1\,dt=2.

    Why: these two numbers, doubled and added, are exactly the identity's right-hand side.

  2. 2

    Form f+g and f-g, then compute \lVert f+g\rVert^2 and \lVert f-g\rVert^2 under that same norm.

    Example

    f+g=t+1 and f-g=t-1, so \lVert f+g\rVert_2^2=\displaystyle\int_{-1}^1(t+1)^2\,dt=\frac83, and by the matching computation \lVert f-g\rVert_2^2=\frac83 as well.

    Why: these two numbers, added together, are exactly the identity's left-hand side, using the identical norm as Step 1.

  3. 3

    Add the two Step 2 numbers for the left side, and double-and-add the two Step 1 numbers for the right side.

    Example

    Left: \frac83+\frac83=\frac{16}3. Right: 2\left(\frac23\right)+2(2)=\frac43+4=\frac{16}3. The two sides match exactly.

    Why: this sum-of-squares equation is the literal statement of the parallelogram law, not an approximation of it.

  4. 4

    A mismatch is decisive: it rules the norm out completely, for every pair of functions, forever. A match on one pair proves nothing beyond that single pair.

    Example

    Under \lVert\cdot\rVert_\infty, take f(t)=1 and g(t)=+1 for t>0, -1 for t<0, both on [-1,1]. Then \lVert f+g\rVert_\infty^2+\lVert f-g\rVert_\infty^2=4+4=8, while 2\lVert f\rVert_\infty^2+2\lVert g\rVert_\infty^2=2+2=4. The mismatch alone rules \lVert\cdot\rVert_\infty out.

    • the two sides disagree — the norm fails the parallelogram law and can never come from any inner product.
    • the two sides agree — this pair is consistent with an inner product, but a general proof is still needed to conclude the law always holds.

    Why: one failing pair is airtight evidence, while one confirming pair only ever confirms itself.

Common mistakes

  • Mixing two different norms across the four quantities, such as computing \lVert f\rVert in L^2 but \lVert f+g\rVert in L^\infty.
  • Forgetting to square each norm before adding; the identity uses squared norms throughout, not the plain norms.
  • Treating one confirming pair as proof the law holds for every pair under that norm.
  • Concluding a norm supports orthogonal projection just because a single parallelogram check happened to succeed.